Theorem bdinex2 17092

Description: Bounded version of inex2 4268. (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 3421	. 2 ⊢ (𝐵 ∩ 𝐴) = (𝐴 ∩ 𝐵)
2		bdinex2.bd	. . 3 ⊢ BOUNDED 𝐵
3		bdinex2.1	. . 3 ⊢ 𝐴 ∈ V
4	2, 3	bdinex1 17091	. 2 ⊢ (𝐴 ∩ 𝐵) ∈ V
5	1, 4	eqeltri 2311	1 ⊢ (𝐵 ∩ 𝐴) ∈ V

This proof depends on syntax axioms:   ∈ wcel 2209  Vcvv 2821   ∩ cin 3219  BOUNDED wbdc 17032

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220  ax-bdsep 17076

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-in 3226  df-bdc 17033

This theorem is used by:  bdssex  17094