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Theorem inopab 5005
 Description: Intersection of two ordered pair class abstractions. (Contributed by NM, 30-Sep-2002.)
Assertion
Ref Expression
inopab
Distinct variable group:   ,
Allowed substitution hints:   (,)   (,)

Proof of Theorem inopab
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relopab 5001 . . 3
2 relin1 4992 . . 3
31, 2ax-mp 8 . 2
4 relopab 5001 . 2
5 sban 2139 . . . 4
6 sban 2139 . . . . 5
76sbbii 1665 . . . 4
8 opelopabsbOLD 4463 . . . . 5
9 opelopabsbOLD 4463 . . . . 5
108, 9anbi12i 679 . . . 4
115, 7, 103bitr4ri 270 . . 3
12 elin 3530 . . 3
13 opelopabsbOLD 4463 . . 3
1411, 12, 133bitr4i 269 . 2
153, 4, 14eqrelriiv 4970 1
 Colors of variables: wff set class Syntax hints:   wa 359   wceq 1652  wsb 1658   wcel 1725   cin 3319  cop 3817  copab 4265   wrel 4883 This theorem is referenced by:  inxp  5007  resopab  5187  fndmin  5837  wemapwe  7654  frgpuplem  15404  ltbwe  16533  opsrtoslem1  16544  pjfval2  16936  lgsquadlem3  21140  dnwech  27123  fgraphopab  27506 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-opab 4267  df-xp 4884  df-rel 4885
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