Objective

I want to open up my basement stairs. This will make it easier to move things in and out.

Current Situation

Thoughts

I suspect I could add a posts to replace the wall. The 4x4 may be a bit small if I remove the wall. 

The house seems to have the stair openings at the same location from first to third floor. From what I have noticed online, it seems feasible to remove the support if I had doubled up the floor joists around the opening with hangers.

Like the following picture without that 2x4.

Problem areas:

Above the wall, I do have 2 2x10s above the wall. However, I only have 1 floor joist that spans the distance from the middle of the house(iron beam) and the foundation.

What I Think I Can Do

Can I get away with laminating a piece of 2x10 to span fill in the missing portion and then add another 2x10 from the been to the wall. All three laminated together. Cut out the cinder block to sit the 2x10s on top. Hangers everywhere. No posts or other supports.

From the Web

This seems wrong.....Floor joists are only supported by that 2x4 wall! Is this is enough? It does appear that they have doubled up their opening joists.

https://www.youtube.com/watch?v=nHpvlUWvj9U

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https://www.diychatroom.com/threads/safe-to-open-basement-stairwell-wall-load-bearing.678487/

I read somewhere in one of these forums that having double 2x10s framing the stairwell, with spans less than 16', made of spruce/fir, and having correct hangings of perpendicular joists, does not require additional support -- hence the wall is not load-bearing. Lack of top-plate, and in some places (the back corner I haven't removed yet) the vertical studs aren't even in contact with the double 2x10s (there for something to nail to vs. structural).

The wall you are taking out looks like a partition. Your floor joists are being supported with hangers by the (2)2x10 trimmer, and that trimmer is supported with a larger hanger by the (2)2x10 header, which in turn is supported by your main girder and the wall on the other side. The size of all these members appears to be adequate. An engineer or a building designer who knows how to calculate gravity loads could confirm that the members are indeed adequate. Nice house.