Theorem opelopab 4774

Description: The law of concretion. Theorem 9.5 of [Quine] p. 61. (Contributed by NM, 16-May-1995.)

Hypotheses

Ref	Expression
opelopab.1
opelopab.2
opelopab.3
opelopab.4

Assertion

Ref	Expression
opelopab

Distinct variable groups:   ,,   ,,   ,,

Proof of Theorem opelopab

Step	Hyp	Ref	Expression
1		opelopab.1	. 2
2		opelopab.2	. 2
3		opelopab.3	. . 3
4		opelopab.4	. . 3
5	3, 4	opelopabg 4770	. 2
6	1, 2, 5	mp2an 672	1

Syntax hints:  ->wi 4  <->wb 184  =wceq 1395  e.wcel 1818  cvv 3109  <.cop 4035  {copab 4509

This theorem is referenced by:  opabid2  5137  dfres2  5331  f1oiso  6247  elopabi  6861  xporderlem  6911  cnlnssadj  26999  areacirclem5  30111  pellexlem3  30767  dicopelval  36904  dih1dimatlem  37056

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