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Theorem opthprc 4736
 Description: Justification theorem for an ordered pair definition that works for any classes, including proper classes. This is a possible definition implied by the footnote in [Jech] p. 78, which says, "The sophisticated reader will not object to our use of a pair of classes." (Contributed by NM, 28-Sep-2003.)
Assertion
Ref Expression
opthprc

Proof of Theorem opthprc
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 eleq2 2344 . . . . 5
2 0ex 4150 . . . . . . . . 9
32snid 3667 . . . . . . . 8
4 opelxp 4719 . . . . . . . 8
53, 4mpbiran2 885 . . . . . . 7
6 opelxp 4719 . . . . . . . 8
7 0nep0 4181 . . . . . . . . . 10
82elsnc 3663 . . . . . . . . . 10
97, 8nemtbir 2534 . . . . . . . . 9
109bianfi 891 . . . . . . . 8
116, 10bitr4i 243 . . . . . . 7
125, 11orbi12i 507 . . . . . 6
13 elun 3316 . . . . . 6
149biorfi 396 . . . . . 6
1512, 13, 143bitr4ri 269 . . . . 5
16 opelxp 4719 . . . . . . . 8
173, 16mpbiran2 885 . . . . . . 7
18 opelxp 4719 . . . . . . . 8
199bianfi 891 . . . . . . . 8
2018, 19bitr4i 243 . . . . . . 7
2117, 20orbi12i 507 . . . . . 6
22 elun 3316 . . . . . 6
239biorfi 396 . . . . . 6
2421, 22, 233bitr4ri 269 . . . . 5
251, 15, 243bitr4g 279 . . . 4
2625eqrdv 2281 . . 3
27 eleq2 2344 . . . . 5
28 opelxp 4719 . . . . . . . 8
29 p0ex 4197 . . . . . . . . . . . 12
3029elsnc 3663 . . . . . . . . . . 11
31 eqcom 2285 . . . . . . . . . . 11
3230, 31bitri 240 . . . . . . . . . 10
337, 32nemtbir 2534 . . . . . . . . 9
3433bianfi 891 . . . . . . . 8
3528, 34bitr4i 243 . . . . . . 7
3629snid 3667 . . . . . . . 8
37 opelxp 4719 . . . . . . . 8
3836, 37mpbiran2 885 . . . . . . 7
3935, 38orbi12i 507 . . . . . 6
40 elun 3316 . . . . . 6
41 biorf 394 . . . . . . 7
4233, 41ax-mp 8 . . . . . 6
4339, 40, 423bitr4ri 269 . . . . 5
44 opelxp 4719 . . . . . . . 8
4533bianfi 891 . . . . . . . 8
4644, 45bitr4i 243 . . . . . . 7
47 opelxp 4719 . . . . . . . 8
4836, 47mpbiran2 885 . . . . . . 7
4946, 48orbi12i 507 . . . . . 6
50 elun 3316 . . . . . 6
51 biorf 394 . . . . . . 7
5233, 51ax-mp 8 . . . . . 6
5349, 50, 523bitr4ri 269 . . . . 5
5427, 43, 533bitr4g 279 . . . 4
5554eqrdv 2281 . . 3
5626, 55jca 518 . 2
57 xpeq1 4703 . . 3
58 xpeq1 4703 . . 3
59 uneq12 3324 . . 3
6057, 58, 59syl2an 463 . 2
6156, 60impbii 180 1
 Colors of variables: wff set class Syntax hints:   wn 3   wb 176   wo 357   wa 358   wceq 1623   wcel 1684   cun 3150  c0 3455  csn 3640  cop 3643   cxp 4687 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-opab 4078  df-xp 4695
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