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Theorem opthprc 4643
 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
StepHypRef Expression
1 eleq2 2314 . . . . 5
2 0ex 4047 . . . . . . . . 9
32snid 3571 . . . . . . . 8
4 opelxp 4626 . . . . . . . 8
53, 4mpbiran2 890 . . . . . . 7
6 opelxp 4626 . . . . . . . 8
7 0nep0 4075 . . . . . . . . . 10
82elsnc 3567 . . . . . . . . . 10
97, 8nemtbir 2500 . . . . . . . . 9
109bianfi 896 . . . . . . . 8
116, 10bitr4i 245 . . . . . . 7
125, 11orbi12i 509 . . . . . 6
13 elun 3226 . . . . . 6
149biorfi 398 . . . . . 6
1512, 13, 143bitr4ri 271 . . . . 5
16 opelxp 4626 . . . . . . . 8
173, 16mpbiran2 890 . . . . . . 7
18 opelxp 4626 . . . . . . . 8
199bianfi 896 . . . . . . . 8
2018, 19bitr4i 245 . . . . . . 7
2117, 20orbi12i 509 . . . . . 6
22 elun 3226 . . . . . 6
239biorfi 398 . . . . . 6
2421, 22, 233bitr4ri 271 . . . . 5
251, 15, 243bitr4g 281 . . . 4
2625eqrdv 2251 . . 3
27 eleq2 2314 . . . . 5
28 opelxp 4626 . . . . . . . 8
29 p0ex 4091 . . . . . . . . . . . 12
3029elsnc 3567 . . . . . . . . . . 11
31 eqcom 2255 . . . . . . . . . . 11
3230, 31bitri 242 . . . . . . . . . 10
337, 32nemtbir 2500 . . . . . . . . 9
3433bianfi 896 . . . . . . . 8
3528, 34bitr4i 245 . . . . . . 7
3629snid 3571 . . . . . . . 8
37 opelxp 4626 . . . . . . . 8
3836, 37mpbiran2 890 . . . . . . 7
3935, 38orbi12i 509 . . . . . 6
40 elun 3226 . . . . . 6
41 biorf 396 . . . . . . 7
4233, 41ax-mp 10 . . . . . 6
4339, 40, 423bitr4ri 271 . . . . 5
44 opelxp 4626 . . . . . . . 8
4533bianfi 896 . . . . . . . 8
4644, 45bitr4i 245 . . . . . . 7
47 opelxp 4626 . . . . . . . 8
4836, 47mpbiran2 890 . . . . . . 7
4946, 48orbi12i 509 . . . . . 6
50 elun 3226 . . . . . 6
51 biorf 396 . . . . . . 7
5233, 51ax-mp 10 . . . . . 6
5349, 50, 523bitr4ri 271 . . . . 5
5427, 43, 533bitr4g 281 . . . 4
5554eqrdv 2251 . . 3
5626, 55jca 520 . 2
57 xpeq1 4610 . . 3
58 xpeq1 4610 . . 3
59 uneq12 3234 . . 3
6057, 58, 59syl2an 465 . 2
6156, 60impbii 182 1
 Colors of variables: wff set class Syntax hints:   wn 5   wb 178   wo 359   wa 360   wceq 1619   wcel 1621   cun 3076  c0 3362  csn 3544  cop 3547   cxp 4578 This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-opab 3975  df-xp 4594
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