Theorem bdinex2 15798

Description: Bounded version of inex2 4178. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
bdinex2.bd	⊢ BOUNDED 𝐵
bdinex2.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
bdinex2	⊢ (𝐵 ∩ 𝐴) ∈ V

Proof of Theorem bdinex2

Step	Hyp	Ref	Expression
1		incom 3364	. 2 ⊢ (𝐵 ∩ 𝐴) = (𝐴 ∩ 𝐵)
2		bdinex2.bd	. . 3 ⊢ BOUNDED 𝐵
3		bdinex2.1	. . 3 ⊢ 𝐴 ∈ V
4	2, 3	bdinex1 15797	. 2 ⊢ (𝐴 ∩ 𝐵) ∈ V
5	1, 4	eqeltri 2277	1 ⊢ (𝐵 ∩ 𝐴) ∈ V

Syntax hints:   ∈ wcel 2175  Vcvv 2771   ∩ cin 3164  BOUNDED wbdc 15738

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186  ax-bdsep 15782

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-v 2773  df-in 3171  df-bdc 15739

This theorem is referenced by:  bdssex  15800