Theorem inopab 5138

Description: Intersection of two ordered pair class abstractions. (Contributed by NM, 30-Sep-2002.)

Assertion

Ref	Expression
inopab

Distinct variable group:   ,

Proof of Theorem inopab

Dummy variables are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relopab 5134	. . 3
2		relin1 5125	. . 3
3	1, 2	ax-mp 5	. 2
4		relopab 5134	. 2
5		sban 2140	. . . 4
6		sban 2140	. . . . 5
7	6	sbbii 1746	. . . 4
8		opelopabsbALT 4761	. . . . 5
9		opelopabsbALT 4761	. . . . 5
10	8, 9	anbi12i 697	. . . 4
11	5, 7, 10	3bitr4ri 278	. . 3
12		elin 3686	. . 3
13		opelopabsbALT 4761	. . 3
14	11, 12, 13	3bitr4i 277	. 2
15	3, 4, 14	eqrelriiv 5102	1

Syntax hints:  /\wa 369  =wceq 1395  [wsb 1739  e.wcel 1818  i^icin 3474  <.cop 4035  {copab 4509  Relwrel 5009

This theorem is referenced by:  inxp  5140  resopab  5325  fndmin  5994  wemapwe  8160  wemapweOLD  8161  frgpuplem  16790  ltbwe  18137  opsrtoslem1  18148  pjfval2  18740  lgsquadlem3  23631  dnwech  30994  fgraphopab  31170  dfiso2  32568

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-ral 2812  df-rex 2813  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  df-xp 5010  df-rel 5011