Theorem elintg 4483

Description: Membership in class intersection, with the sethood requirement expressed as an antecedent. (Contributed by NM, 20-Nov-2003.) (Proof shortened by JJ, 26-Jul-2021.)

Assertion

Ref	Expression
elintg	|- ( A e. V -> ( A e. |^| B <-> A. x e. B A e. x ) )

Distinct variable groups:   x, A   x, B

Allowed substitution hint:   V( x)

Proof of Theorem elintg

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2689	. . 3 |- ( y = A -> ( y e. x <-> A e. x ) )
2	1	ralbidv 2986	. 2 |- ( y = A -> ( A. x e. B y e. x <-> A. x e. B A e. x ) )
3		dfint2 4477	. 2 |- |^| B = { y | A. x e. B y e. x }
4	2, 3	elab2g 3353	1 |- ( A e. V -> ( A e. |^| B <-> A. x e. B A e. x ) )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   A.wral 2912   |^|cint 4475

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-v 3202  df-int 4476

This theorem is referenced by:  elinti  4485  elrint  4518  onmindif  5815  onmindif2  7012  mremre  16264  toponmre  20897  1stcfb  21248  uffixfr  21727  plycpn  24044  insiga  30200  dfon2lem8  31695  elintabg  37880  trintALTVD  39116  trintALT  39117  elintd  39245  intsaluni  40547  intsal  40548