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Theorem opthprc 4887
 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 2538 . . . . 5
2 0ex 4528 . . . . . . . . 9
32snid 3988 . . . . . . . 8
4 opelxp 4869 . . . . . . . 8
53, 4mpbiran2 933 . . . . . . 7
6 opelxp 4869 . . . . . . . 8
7 0nep0 4572 . . . . . . . . . 10
82elsnc 3984 . . . . . . . . . 10
97, 8nemtbir 2738 . . . . . . . . 9
109bianfi 939 . . . . . . . 8
116, 10bitr4i 260 . . . . . . 7
125, 11orbi12i 530 . . . . . 6
13 elun 3565 . . . . . 6
149biorfi 414 . . . . . 6
1512, 13, 143bitr4ri 286 . . . . 5
16 opelxp 4869 . . . . . . . 8
173, 16mpbiran2 933 . . . . . . 7
18 opelxp 4869 . . . . . . . 8
199bianfi 939 . . . . . . . 8
2018, 19bitr4i 260 . . . . . . 7
2117, 20orbi12i 530 . . . . . 6
22 elun 3565 . . . . . 6
239biorfi 414 . . . . . 6
2421, 22, 233bitr4ri 286 . . . . 5
251, 15, 243bitr4g 296 . . . 4
2625eqrdv 2469 . . 3
27 eleq2 2538 . . . . 5
28 opelxp 4869 . . . . . . . 8
29 p0ex 4588 . . . . . . . . . . . 12
3029elsnc 3984 . . . . . . . . . . 11
31 eqcom 2478 . . . . . . . . . . 11
3230, 31bitri 257 . . . . . . . . . 10
337, 32nemtbir 2738 . . . . . . . . 9
3433bianfi 939 . . . . . . . 8
3528, 34bitr4i 260 . . . . . . 7
3629snid 3988 . . . . . . . 8
37 opelxp 4869 . . . . . . . 8
3836, 37mpbiran2 933 . . . . . . 7
3935, 38orbi12i 530 . . . . . 6
40 elun 3565 . . . . . 6
41 biorf 412 . . . . . . 7
4233, 41ax-mp 5 . . . . . 6
4339, 40, 423bitr4ri 286 . . . . 5
44 opelxp 4869 . . . . . . . 8
4533bianfi 939 . . . . . . . 8
4644, 45bitr4i 260 . . . . . . 7
47 opelxp 4869 . . . . . . . 8
4836, 47mpbiran2 933 . . . . . . 7
4946, 48orbi12i 530 . . . . . 6
50 elun 3565 . . . . . 6
51 biorf 412 . . . . . . 7
5233, 51ax-mp 5 . . . . . 6
5349, 50, 523bitr4ri 286 . . . . 5
5427, 43, 533bitr4g 296 . . . 4
5554eqrdv 2469 . . 3
5626, 55jca 541 . 2
57 xpeq1 4853 . . 3
58 xpeq1 4853 . . 3
59 uneq12 3574 . . 3
6057, 58, 59syl2an 485 . 2
6156, 60impbii 192 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wb 189   wo 375   wa 376   wceq 1452   wcel 1904   cun 3388  c0 3722  csn 3959  cop 3965   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-pow 4579  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-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-pw 3944  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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