Theorem bdinex2 16035

Description: Bounded version of inex2 4195. (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 3373	. 2 ⊢ (𝐵 ∩ 𝐴) = (𝐴 ∩ 𝐵)
2		bdinex2.bd	. . 3 ⊢ BOUNDED 𝐵
3		bdinex2.1	. . 3 ⊢ 𝐴 ∈ V
4	2, 3	bdinex1 16034	. 2 ⊢ (𝐴 ∩ 𝐵) ∈ V
5	1, 4	eqeltri 2280	1 ⊢ (𝐵 ∩ 𝐴) ∈ V

Syntax hints:   ∈ wcel 2178  Vcvv 2776   ∩ cin 3173  BOUNDED wbdc 15975

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189  ax-bdsep 16019

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-in 3180  df-bdc 15976

This theorem is referenced by:  bdssex  16037