Theorem elopab 4760

Description: Membership in a class abstraction of pairs. (Contributed by NM, 24-Mar-1998.)

Assertion

Ref	Expression
elopab

Distinct variable groups:   ,   ,

Proof of Theorem elopab

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3118	. 2
2		opex 4716	. . . . 5
3		eleq1 2529	. . . . 5
4	2, 3	mpbiri 233	. . . 4
5	4	adantr 465	. . 3
6	5	exlimivv 1723	. 2
7		eqeq1 2461	. . . . 5
8	7	anbi1d 704	. . . 4
9	8	2exbidv 1716	. . 3
10		df-opab 4511	. . 3
11	9, 10	elab2g 3248	. 2
12	1, 6, 11	pm5.21nii 353	1

Syntax hints:  <->wb 184  /\wa 369  =wceq 1395  E.wex 1612  e.wcel 1818  cvv 3109  <.cop 4035  {copab 4509

This theorem is referenced by:  opelopabsbALT  4761  opelopabsb  4762  opelopabt  4764  opelopabga  4765  opabn0  4783  iunopab  4788  epelg  4797  elxp  5021  elopaelxp  5077  elopaba  5120  elcnv  5184  dfmpt3  5708  fmptsng  6092  fmptsnd  6093  0neqopab  6341  opabex3d  6778  opabex3  6779  fsplit  6905  isfunc  15233  usgraop  24350  clwlkswlks  24758  qqhval2  27963  eulerpartlemgvv  28315  rtrclreclem.trans  29069  pellexlem5  30769  pellex  30771  opelopab4  33324  dicelval3  36907

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-opab 4511