Theorem bdinex2 15546

Description: Bounded version of inex2 4168. (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 3355	. 2 ⊢ (𝐵 ∩ 𝐴) = (𝐴 ∩ 𝐵)
2		bdinex2.bd	. . 3 ⊢ BOUNDED 𝐵
3		bdinex2.1	. . 3 ⊢ 𝐴 ∈ V
4	2, 3	bdinex1 15545	. 2 ⊢ (𝐴 ∩ 𝐵) ∈ V
5	1, 4	eqeltri 2269	1 ⊢ (𝐵 ∩ 𝐴) ∈ V

Syntax hints:   ∈ wcel 2167  Vcvv 2763   ∩ cin 3156  BOUNDED wbdc 15486

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178  ax-bdsep 15530

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-in 3163  df-bdc 15487

This theorem is referenced by:  bdssex  15548