Theorem inex2 5265

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 4163	. 2 ⊢ (𝐵 ∩ 𝐴) = (𝐴 ∩ 𝐵)
2		inex2.1	. . 3 ⊢ 𝐴 ∈ V
3	2	inex1 5264	. 2 ⊢ (𝐴 ∩ 𝐵) ∈ V
4	1, 3	eqeltri 2833	1 ⊢ (𝐵 ∩ 𝐴) ∈ V

Syntax hints:   ∈ wcel 2114  Vcvv 3442   ∩ cin 3902

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 5243

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 3402  df-v 3444  df-in 3910

This theorem is referenced by:  ssex  5268  wefrc  5626  hartogslem1  9459  infxpenlem  9935  dfac5lem5  10049  fin23lem12  10253  fpwwe2lem11  10564  cnso  16184  ressbas  17175  ressress  17186  rescabs  17769  symgvalstruct  19338  mgpress  20097  pjfval  21673  tgdom  22934  distop  22951  ustfilxp  24169  elovolmlem  25443  dyadmbl  25569  volsup2  25574  vitali  25582  itg1climres  25683  tayl0  26337  atomli  32470  ldgenpisyslem1  34341  reprinfz1  34800  bj-elid4  37423  aomclem6  43416  elinintrab  43933  isotone2  44405  ntrrn  44478  ntrf  44479  dssmapntrcls  44484  ismnushort  44657  onfrALTlem3  44900  sswfaxreg  45343  limcresiooub  46000  limcresioolb  46001  limsupval4  46152  sge0iunmptlemre  46773  ovolval2lem  47001  ovolval4lem2  47008  nthrucw  47244  setrec2fun  50051