Theorem bdinex2 13782

Description: Bounded version of inex2 4117. (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 3314	. 2 ⊢ (𝐵 ∩ 𝐴) = (𝐴 ∩ 𝐵)
2		bdinex2.bd	. . 3 ⊢ BOUNDED 𝐵
3		bdinex2.1	. . 3 ⊢ 𝐴 ∈ V
4	2, 3	bdinex1 13781	. 2 ⊢ (𝐴 ∩ 𝐵) ∈ V
5	1, 4	eqeltri 2239	1 ⊢ (𝐵 ∩ 𝐴) ∈ V

Syntax hints:   ∈ wcel 2136  Vcvv 2726   ∩ cin 3115  BOUNDED wbdc 13722

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147  ax-bdsep 13766

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-in 3122  df-bdc 13723

This theorem is referenced by:  bdssex  13784