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Theorem opthprc 5201
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 (((𝐴 × {∅}) ∪ (𝐵 × {{∅}})) = ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})) ↔ (𝐴 = 𝐶𝐵 = 𝐷))

Proof of Theorem opthprc
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eleq2 2719 . . . . 5 (((𝐴 × {∅}) ∪ (𝐵 × {{∅}})) = ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})) → (⟨𝑥, ∅⟩ ∈ ((𝐴 × {∅}) ∪ (𝐵 × {{∅}})) ↔ ⟨𝑥, ∅⟩ ∈ ((𝐶 × {∅}) ∪ (𝐷 × {{∅}}))))
2 0ex 4823 . . . . . . . . 9 ∅ ∈ V
32snid 4241 . . . . . . . 8 ∅ ∈ {∅}
4 opelxp 5180 . . . . . . . 8 (⟨𝑥, ∅⟩ ∈ (𝐴 × {∅}) ↔ (𝑥𝐴 ∧ ∅ ∈ {∅}))
53, 4mpbiran2 974 . . . . . . 7 (⟨𝑥, ∅⟩ ∈ (𝐴 × {∅}) ↔ 𝑥𝐴)
6 opelxp 5180 . . . . . . . 8 (⟨𝑥, ∅⟩ ∈ (𝐵 × {{∅}}) ↔ (𝑥𝐵 ∧ ∅ ∈ {{∅}}))
7 0nep0 4866 . . . . . . . . . 10 ∅ ≠ {∅}
82elsn 4225 . . . . . . . . . 10 (∅ ∈ {{∅}} ↔ ∅ = {∅})
97, 8nemtbir 2918 . . . . . . . . 9 ¬ ∅ ∈ {{∅}}
109bianfi 986 . . . . . . . 8 (∅ ∈ {{∅}} ↔ (𝑥𝐵 ∧ ∅ ∈ {{∅}}))
116, 10bitr4i 267 . . . . . . 7 (⟨𝑥, ∅⟩ ∈ (𝐵 × {{∅}}) ↔ ∅ ∈ {{∅}})
125, 11orbi12i 542 . . . . . 6 ((⟨𝑥, ∅⟩ ∈ (𝐴 × {∅}) ∨ ⟨𝑥, ∅⟩ ∈ (𝐵 × {{∅}})) ↔ (𝑥𝐴 ∨ ∅ ∈ {{∅}}))
13 elun 3786 . . . . . 6 (⟨𝑥, ∅⟩ ∈ ((𝐴 × {∅}) ∪ (𝐵 × {{∅}})) ↔ (⟨𝑥, ∅⟩ ∈ (𝐴 × {∅}) ∨ ⟨𝑥, ∅⟩ ∈ (𝐵 × {{∅}})))
149biorfi 421 . . . . . 6 (𝑥𝐴 ↔ (𝑥𝐴 ∨ ∅ ∈ {{∅}}))
1512, 13, 143bitr4ri 293 . . . . 5 (𝑥𝐴 ↔ ⟨𝑥, ∅⟩ ∈ ((𝐴 × {∅}) ∪ (𝐵 × {{∅}})))
16 opelxp 5180 . . . . . . . 8 (⟨𝑥, ∅⟩ ∈ (𝐶 × {∅}) ↔ (𝑥𝐶 ∧ ∅ ∈ {∅}))
173, 16mpbiran2 974 . . . . . . 7 (⟨𝑥, ∅⟩ ∈ (𝐶 × {∅}) ↔ 𝑥𝐶)
18 opelxp 5180 . . . . . . . 8 (⟨𝑥, ∅⟩ ∈ (𝐷 × {{∅}}) ↔ (𝑥𝐷 ∧ ∅ ∈ {{∅}}))
199bianfi 986 . . . . . . . 8 (∅ ∈ {{∅}} ↔ (𝑥𝐷 ∧ ∅ ∈ {{∅}}))
2018, 19bitr4i 267 . . . . . . 7 (⟨𝑥, ∅⟩ ∈ (𝐷 × {{∅}}) ↔ ∅ ∈ {{∅}})
2117, 20orbi12i 542 . . . . . 6 ((⟨𝑥, ∅⟩ ∈ (𝐶 × {∅}) ∨ ⟨𝑥, ∅⟩ ∈ (𝐷 × {{∅}})) ↔ (𝑥𝐶 ∨ ∅ ∈ {{∅}}))
22 elun 3786 . . . . . 6 (⟨𝑥, ∅⟩ ∈ ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})) ↔ (⟨𝑥, ∅⟩ ∈ (𝐶 × {∅}) ∨ ⟨𝑥, ∅⟩ ∈ (𝐷 × {{∅}})))
239biorfi 421 . . . . . 6 (𝑥𝐶 ↔ (𝑥𝐶 ∨ ∅ ∈ {{∅}}))
2421, 22, 233bitr4ri 293 . . . . 5 (𝑥𝐶 ↔ ⟨𝑥, ∅⟩ ∈ ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})))
251, 15, 243bitr4g 303 . . . 4 (((𝐴 × {∅}) ∪ (𝐵 × {{∅}})) = ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})) → (𝑥𝐴𝑥𝐶))
2625eqrdv 2649 . . 3 (((𝐴 × {∅}) ∪ (𝐵 × {{∅}})) = ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})) → 𝐴 = 𝐶)
27 eleq2 2719 . . . . 5 (((𝐴 × {∅}) ∪ (𝐵 × {{∅}})) = ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})) → (⟨𝑥, {∅}⟩ ∈ ((𝐴 × {∅}) ∪ (𝐵 × {{∅}})) ↔ ⟨𝑥, {∅}⟩ ∈ ((𝐶 × {∅}) ∪ (𝐷 × {{∅}}))))
28 opelxp 5180 . . . . . . . 8 (⟨𝑥, {∅}⟩ ∈ (𝐴 × {∅}) ↔ (𝑥𝐴 ∧ {∅} ∈ {∅}))
29 p0ex 4883 . . . . . . . . . . . 12 {∅} ∈ V
3029elsn 4225 . . . . . . . . . . 11 ({∅} ∈ {∅} ↔ {∅} = ∅)
31 eqcom 2658 . . . . . . . . . . 11 ({∅} = ∅ ↔ ∅ = {∅})
3230, 31bitri 264 . . . . . . . . . 10 ({∅} ∈ {∅} ↔ ∅ = {∅})
337, 32nemtbir 2918 . . . . . . . . 9 ¬ {∅} ∈ {∅}
3433bianfi 986 . . . . . . . 8 ({∅} ∈ {∅} ↔ (𝑥𝐴 ∧ {∅} ∈ {∅}))
3528, 34bitr4i 267 . . . . . . 7 (⟨𝑥, {∅}⟩ ∈ (𝐴 × {∅}) ↔ {∅} ∈ {∅})
3629snid 4241 . . . . . . . 8 {∅} ∈ {{∅}}
37 opelxp 5180 . . . . . . . 8 (⟨𝑥, {∅}⟩ ∈ (𝐵 × {{∅}}) ↔ (𝑥𝐵 ∧ {∅} ∈ {{∅}}))
3836, 37mpbiran2 974 . . . . . . 7 (⟨𝑥, {∅}⟩ ∈ (𝐵 × {{∅}}) ↔ 𝑥𝐵)
3935, 38orbi12i 542 . . . . . 6 ((⟨𝑥, {∅}⟩ ∈ (𝐴 × {∅}) ∨ ⟨𝑥, {∅}⟩ ∈ (𝐵 × {{∅}})) ↔ ({∅} ∈ {∅} ∨ 𝑥𝐵))
40 elun 3786 . . . . . 6 (⟨𝑥, {∅}⟩ ∈ ((𝐴 × {∅}) ∪ (𝐵 × {{∅}})) ↔ (⟨𝑥, {∅}⟩ ∈ (𝐴 × {∅}) ∨ ⟨𝑥, {∅}⟩ ∈ (𝐵 × {{∅}})))
41 biorf 419 . . . . . . 7 (¬ {∅} ∈ {∅} → (𝑥𝐵 ↔ ({∅} ∈ {∅} ∨ 𝑥𝐵)))
4233, 41ax-mp 5 . . . . . 6 (𝑥𝐵 ↔ ({∅} ∈ {∅} ∨ 𝑥𝐵))
4339, 40, 423bitr4ri 293 . . . . 5 (𝑥𝐵 ↔ ⟨𝑥, {∅}⟩ ∈ ((𝐴 × {∅}) ∪ (𝐵 × {{∅}})))
44 opelxp 5180 . . . . . . . 8 (⟨𝑥, {∅}⟩ ∈ (𝐶 × {∅}) ↔ (𝑥𝐶 ∧ {∅} ∈ {∅}))
4533bianfi 986 . . . . . . . 8 ({∅} ∈ {∅} ↔ (𝑥𝐶 ∧ {∅} ∈ {∅}))
4644, 45bitr4i 267 . . . . . . 7 (⟨𝑥, {∅}⟩ ∈ (𝐶 × {∅}) ↔ {∅} ∈ {∅})
47 opelxp 5180 . . . . . . . 8 (⟨𝑥, {∅}⟩ ∈ (𝐷 × {{∅}}) ↔ (𝑥𝐷 ∧ {∅} ∈ {{∅}}))
4836, 47mpbiran2 974 . . . . . . 7 (⟨𝑥, {∅}⟩ ∈ (𝐷 × {{∅}}) ↔ 𝑥𝐷)
4946, 48orbi12i 542 . . . . . 6 ((⟨𝑥, {∅}⟩ ∈ (𝐶 × {∅}) ∨ ⟨𝑥, {∅}⟩ ∈ (𝐷 × {{∅}})) ↔ ({∅} ∈ {∅} ∨ 𝑥𝐷))
50 elun 3786 . . . . . 6 (⟨𝑥, {∅}⟩ ∈ ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})) ↔ (⟨𝑥, {∅}⟩ ∈ (𝐶 × {∅}) ∨ ⟨𝑥, {∅}⟩ ∈ (𝐷 × {{∅}})))
51 biorf 419 . . . . . . 7 (¬ {∅} ∈ {∅} → (𝑥𝐷 ↔ ({∅} ∈ {∅} ∨ 𝑥𝐷)))
5233, 51ax-mp 5 . . . . . 6 (𝑥𝐷 ↔ ({∅} ∈ {∅} ∨ 𝑥𝐷))
5349, 50, 523bitr4ri 293 . . . . 5 (𝑥𝐷 ↔ ⟨𝑥, {∅}⟩ ∈ ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})))
5427, 43, 533bitr4g 303 . . . 4 (((𝐴 × {∅}) ∪ (𝐵 × {{∅}})) = ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})) → (𝑥𝐵𝑥𝐷))
5554eqrdv 2649 . . 3 (((𝐴 × {∅}) ∪ (𝐵 × {{∅}})) = ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})) → 𝐵 = 𝐷)
5626, 55jca 553 . 2 (((𝐴 × {∅}) ∪ (𝐵 × {{∅}})) = ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})) → (𝐴 = 𝐶𝐵 = 𝐷))
57 xpeq1 5157 . . 3 (𝐴 = 𝐶 → (𝐴 × {∅}) = (𝐶 × {∅}))
58 xpeq1 5157 . . 3 (𝐵 = 𝐷 → (𝐵 × {{∅}}) = (𝐷 × {{∅}}))
59 uneq12 3795 . . 3 (((𝐴 × {∅}) = (𝐶 × {∅}) ∧ (𝐵 × {{∅}}) = (𝐷 × {{∅}})) → ((𝐴 × {∅}) ∪ (𝐵 × {{∅}})) = ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})))
6057, 58, 59syl2an 493 . 2 ((𝐴 = 𝐶𝐵 = 𝐷) → ((𝐴 × {∅}) ∪ (𝐵 × {{∅}})) = ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})))
6156, 60impbii 199 1 (((𝐴 × {∅}) ∪ (𝐵 × {{∅}})) = ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})) ↔ (𝐴 = 𝐶𝐵 = 𝐷))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wo 382  wa 383   = wceq 1523  wcel 2030  cun 3605  c0 3948  {csn 4210  cop 4216   × cxp 5141
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-opab 4746  df-xp 5149
This theorem is referenced by: (None)
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