Theorem opelopabga 4765

Description: The law of concretion. Theorem 9.5 of [Quine] p. 61. (Contributed by Mario Carneiro, 19-Dec-2013.)

Hypothesis

Ref	Expression
opelopabga.1

Assertion

Ref	Expression
opelopabga

Distinct variable groups:   ,,   ,,   ,,

Proof of Theorem opelopabga

Step	Hyp	Ref	Expression
1		elopab 4760	. 2
2		opelopabga.1	. . 3
3	2	copsex2g 4740	. 2
4	1, 3	syl5bb 257	1

Syntax hints:  ->wi 4  <->wb 184  /\wa 369  =wceq 1395  E.wex 1612  e.wcel 1818  <.cop 4035  {copab 4509

This theorem is referenced by:  brabga  4766  opelopab2a  4767  opelopaba  4768  opelopabg  4770  fmptsng  6092  isprmpt2  6972  canthwelem  9049  iswlk  24520  istrl  24539  ispth  24570  isspth  24571  isclwlk0  24754  isrngo  25380

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-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-rab 2816  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