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Theorem opelopab3 28759
 Description: Ordered pair membership in an ordered pair class abstraction, with a reduced hypothesis. (Contributed by Jeff Madsen, 29-May-2011.)
Hypotheses
Ref Expression
opelopab3.1
opelopab3.2
opelopab3.3
Assertion
Ref Expression
opelopab3
Distinct variable groups:   ,,   ,,   ,,
Allowed substitution hints:   (,)   (,)   (,)   (,)

Proof of Theorem opelopab3
StepHypRef Expression
1 relopab 5075 . . . . . . 7
2 df-rel 4956 . . . . . . 7
31, 2mpbi 208 . . . . . 6
43sseli 3461 . . . . 5
5 opelxp1 4981 . . . . 5
64, 5syl 16 . . . 4
76anim1i 568 . . 3
87ancoms 453 . 2
9 opelopab3.3 . . . . 5
10 elex 3087 . . . . 5
119, 10syl 16 . . . 4
1211anim1i 568 . . 3
1312ancoms 453 . 2
14 opelopab3.1 . . 3
15 opelopab3.2 . . 3
1614, 15opelopabg 4716 . 2
178, 13, 16pm5.21nd 893 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184   wa 369   wceq 1370   wcel 1758  cvv 3078   wss 3437  cop 3992  copab 4458   cxp 4947   wrel 4954 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pr 4640 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-op 3993  df-opab 4460  df-xp 4955  df-rel 4956 This theorem is referenced by: (None)
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