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Theorem opelopab3 32107
 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 elopaelxp 4912 . . . . 5
2 opelxp1 4872 . . . . 5
31, 2syl 17 . . . 4
43anim1i 578 . . 3
54ancoms 460 . 2
6 opelopab3.3 . . . . 5
7 elex 3040 . . . . 5
86, 7syl 17 . . . 4
98anim1i 578 . . 3
109ancoms 460 . 2
11 opelopab3.1 . . 3
12 opelopab3.2 . . 3
1311, 12opelopabg 4719 . 2
145, 10, 13pm5.21nd 916 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 189   wa 376   wceq 1452   wcel 1904  cvv 3031  cop 3965  copab 4453   cxp 4837 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639 This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-opab 4455  df-xp 4845 This theorem is referenced by: (None)
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