Theorem inex2 5256

Description: Separation Scheme (Aussonderung) using class notation. (Contributed by NM, 27-Apr-1994.)

Hypothesis

Ref	Expression
inex2.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
inex2	⊢ (𝐵 ∩ 𝐴) ∈ V

Proof of Theorem inex2

Step	Hyp	Ref	Expression
1		incom 4150	. 2 ⊢ (𝐵 ∩ 𝐴) = (𝐴 ∩ 𝐵)
2		inex2.1	. . 3 ⊢ 𝐴 ∈ V
3	2	inex1 5255	. 2 ⊢ (𝐴 ∩ 𝐵) ∈ V
4	1, 3	eqeltri 2833	1 ⊢ (𝐵 ∩ 𝐴) ∈ V

Syntax hints:   ∈ wcel 2114  Vcvv 3430   ∩ cin 3889

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5232

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3391  df-v 3432  df-in 3897

This theorem is referenced by:  ssex  5259  wefrc  5619  hartogslem1  9451  infxpenlem  9929  dfac5lem5  10043  fin23lem12  10247  fpwwe2lem11  10558  cnso  16208  ressbas  17200  ressress  17211  rescabs  17794  symgvalstruct  19366  mgpress  20125  pjfval  21699  tgdom  22956  distop  22973  ustfilxp  24191  elovolmlem  25454  dyadmbl  25580  volsup2  25585  vitali  25593  itg1climres  25694  tayl0  26341  atomli  32471  ldgenpisyslem1  34326  reprinfz1  34785  dfttc4  36731  bj-elid4  37501  aomclem6  43508  elinintrab  44025  isotone2  44497  ntrrn  44570  ntrf  44571  dssmapntrcls  44576  ismnushort  44749  onfrALTlem3  44992  sswfaxreg  45435  limcresiooub  46091  limcresioolb  46092  limsupval4  46243  sge0iunmptlemre  46864  ovolval2lem  47092  ovolval4lem2  47099  nthrucw  47335  setrec2fun  50182