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Theorem inopab 4583
 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 4579 . . 3
2 relin1 4570 . . 3
31, 2ax-mp 7 . 2
4 relopab 4579 . 2
5 sban 1878 . . . 4
6 sban 1878 . . . . 5
76sbbii 1696 . . . 4
8 opelopabsbALT 4097 . . . . 5
9 opelopabsbALT 4097 . . . . 5
108, 9anbi12i 449 . . . 4
115, 7, 103bitr4ri 212 . . 3
12 elin 3186 . . 3
13 opelopabsbALT 4097 . . 3
1411, 12, 133bitr4i 211 . 2
153, 4, 14eqrelriiv 4547 1
 Colors of variables: wff set class Syntax hints:   wa 103   wceq 1290   wcel 1439  wsb 1693   cin 3001  cop 3455  copab 3906   wrel 4459 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3965  ax-pow 4017  ax-pr 4047 This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2624  df-un 3006  df-in 3008  df-ss 3015  df-pw 3437  df-sn 3458  df-pr 3459  df-op 3461  df-opab 3908  df-xp 4460  df-rel 4461 This theorem is referenced by:  inxp  4585  resopab  4771  cnvin  4854  fndmin  5422  enq0enq  7053
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