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Theorem opelopab3 26409
 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 4993 . . . . . . 7
2 df-rel 4877 . . . . . . 7
31, 2mpbi 200 . . . . . 6
43sseli 3336 . . . . 5
5 opelxp1 4903 . . . . 5
64, 5syl 16 . . . 4
76anim1i 552 . . 3
87ancoms 440 . 2
9 opelopab3.3 . . . . 5
10 elex 2956 . . . . 5
119, 10syl 16 . . . 4
1211anim1i 552 . . 3
1312ancoms 440 . 2
14 opelopab3.1 . . 3
15 opelopab3.2 . . 3
1614, 15opelopabg 4465 . 2
178, 13, 16pm5.21nd 869 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359   wceq 1652   wcel 1725  cvv 2948   wss 3312  cop 3809  copab 4257   cxp 4868   wrel 4875 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 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395 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-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-opab 4259  df-xp 4876  df-rel 4877
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