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Theorem opthprc 5157
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 2688 . . . . 5 (((𝐴 × {∅}) ∪ (𝐵 × {{∅}})) = ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})) → (⟨𝑥, ∅⟩ ∈ ((𝐴 × {∅}) ∪ (𝐵 × {{∅}})) ↔ ⟨𝑥, ∅⟩ ∈ ((𝐶 × {∅}) ∪ (𝐷 × {{∅}}))))
2 0ex 4781 . . . . . . . . 9 ∅ ∈ V
32snid 4199 . . . . . . . 8 ∅ ∈ {∅}
4 opelxp 5136 . . . . . . . 8 (⟨𝑥, ∅⟩ ∈ (𝐴 × {∅}) ↔ (𝑥𝐴 ∧ ∅ ∈ {∅}))
53, 4mpbiran2 953 . . . . . . 7 (⟨𝑥, ∅⟩ ∈ (𝐴 × {∅}) ↔ 𝑥𝐴)
6 opelxp 5136 . . . . . . . 8 (⟨𝑥, ∅⟩ ∈ (𝐵 × {{∅}}) ↔ (𝑥𝐵 ∧ ∅ ∈ {{∅}}))
7 0nep0 4827 . . . . . . . . . 10 ∅ ≠ {∅}
82elsn 4183 . . . . . . . . . 10 (∅ ∈ {{∅}} ↔ ∅ = {∅})
97, 8nemtbir 2886 . . . . . . . . 9 ¬ ∅ ∈ {{∅}}
109bianfi 965 . . . . . . . 8 (∅ ∈ {{∅}} ↔ (𝑥𝐵 ∧ ∅ ∈ {{∅}}))
116, 10bitr4i 267 . . . . . . 7 (⟨𝑥, ∅⟩ ∈ (𝐵 × {{∅}}) ↔ ∅ ∈ {{∅}})
125, 11orbi12i 543 . . . . . 6 ((⟨𝑥, ∅⟩ ∈ (𝐴 × {∅}) ∨ ⟨𝑥, ∅⟩ ∈ (𝐵 × {{∅}})) ↔ (𝑥𝐴 ∨ ∅ ∈ {{∅}}))
13 elun 3745 . . . . . 6 (⟨𝑥, ∅⟩ ∈ ((𝐴 × {∅}) ∪ (𝐵 × {{∅}})) ↔ (⟨𝑥, ∅⟩ ∈ (𝐴 × {∅}) ∨ ⟨𝑥, ∅⟩ ∈ (𝐵 × {{∅}})))
149biorfi 422 . . . . . 6 (𝑥𝐴 ↔ (𝑥𝐴 ∨ ∅ ∈ {{∅}}))
1512, 13, 143bitr4ri 293 . . . . 5 (𝑥𝐴 ↔ ⟨𝑥, ∅⟩ ∈ ((𝐴 × {∅}) ∪ (𝐵 × {{∅}})))
16 opelxp 5136 . . . . . . . 8 (⟨𝑥, ∅⟩ ∈ (𝐶 × {∅}) ↔ (𝑥𝐶 ∧ ∅ ∈ {∅}))
173, 16mpbiran2 953 . . . . . . 7 (⟨𝑥, ∅⟩ ∈ (𝐶 × {∅}) ↔ 𝑥𝐶)
18 opelxp 5136 . . . . . . . 8 (⟨𝑥, ∅⟩ ∈ (𝐷 × {{∅}}) ↔ (𝑥𝐷 ∧ ∅ ∈ {{∅}}))
199bianfi 965 . . . . . . . 8 (∅ ∈ {{∅}} ↔ (𝑥𝐷 ∧ ∅ ∈ {{∅}}))
2018, 19bitr4i 267 . . . . . . 7 (⟨𝑥, ∅⟩ ∈ (𝐷 × {{∅}}) ↔ ∅ ∈ {{∅}})
2117, 20orbi12i 543 . . . . . 6 ((⟨𝑥, ∅⟩ ∈ (𝐶 × {∅}) ∨ ⟨𝑥, ∅⟩ ∈ (𝐷 × {{∅}})) ↔ (𝑥𝐶 ∨ ∅ ∈ {{∅}}))
22 elun 3745 . . . . . 6 (⟨𝑥, ∅⟩ ∈ ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})) ↔ (⟨𝑥, ∅⟩ ∈ (𝐶 × {∅}) ∨ ⟨𝑥, ∅⟩ ∈ (𝐷 × {{∅}})))
239biorfi 422 . . . . . 6 (𝑥𝐶 ↔ (𝑥𝐶 ∨ ∅ ∈ {{∅}}))
2421, 22, 233bitr4ri 293 . . . . 5 (𝑥𝐶 ↔ ⟨𝑥, ∅⟩ ∈ ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})))
251, 15, 243bitr4g 303 . . . 4 (((𝐴 × {∅}) ∪ (𝐵 × {{∅}})) = ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})) → (𝑥𝐴𝑥𝐶))
2625eqrdv 2618 . . 3 (((𝐴 × {∅}) ∪ (𝐵 × {{∅}})) = ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})) → 𝐴 = 𝐶)
27 eleq2 2688 . . . . 5 (((𝐴 × {∅}) ∪ (𝐵 × {{∅}})) = ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})) → (⟨𝑥, {∅}⟩ ∈ ((𝐴 × {∅}) ∪ (𝐵 × {{∅}})) ↔ ⟨𝑥, {∅}⟩ ∈ ((𝐶 × {∅}) ∪ (𝐷 × {{∅}}))))
28 opelxp 5136 . . . . . . . 8 (⟨𝑥, {∅}⟩ ∈ (𝐴 × {∅}) ↔ (𝑥𝐴 ∧ {∅} ∈ {∅}))
29 p0ex 4844 . . . . . . . . . . . 12 {∅} ∈ V
3029elsn 4183 . . . . . . . . . . 11 ({∅} ∈ {∅} ↔ {∅} = ∅)
31 eqcom 2627 . . . . . . . . . . 11 ({∅} = ∅ ↔ ∅ = {∅})
3230, 31bitri 264 . . . . . . . . . 10 ({∅} ∈ {∅} ↔ ∅ = {∅})
337, 32nemtbir 2886 . . . . . . . . 9 ¬ {∅} ∈ {∅}
3433bianfi 965 . . . . . . . 8 ({∅} ∈ {∅} ↔ (𝑥𝐴 ∧ {∅} ∈ {∅}))
3528, 34bitr4i 267 . . . . . . 7 (⟨𝑥, {∅}⟩ ∈ (𝐴 × {∅}) ↔ {∅} ∈ {∅})
3629snid 4199 . . . . . . . 8 {∅} ∈ {{∅}}
37 opelxp 5136 . . . . . . . 8 (⟨𝑥, {∅}⟩ ∈ (𝐵 × {{∅}}) ↔ (𝑥𝐵 ∧ {∅} ∈ {{∅}}))
3836, 37mpbiran2 953 . . . . . . 7 (⟨𝑥, {∅}⟩ ∈ (𝐵 × {{∅}}) ↔ 𝑥𝐵)
3935, 38orbi12i 543 . . . . . 6 ((⟨𝑥, {∅}⟩ ∈ (𝐴 × {∅}) ∨ ⟨𝑥, {∅}⟩ ∈ (𝐵 × {{∅}})) ↔ ({∅} ∈ {∅} ∨ 𝑥𝐵))
40 elun 3745 . . . . . 6 (⟨𝑥, {∅}⟩ ∈ ((𝐴 × {∅}) ∪ (𝐵 × {{∅}})) ↔ (⟨𝑥, {∅}⟩ ∈ (𝐴 × {∅}) ∨ ⟨𝑥, {∅}⟩ ∈ (𝐵 × {{∅}})))
41 biorf 420 . . . . . . 7 (¬ {∅} ∈ {∅} → (𝑥𝐵 ↔ ({∅} ∈ {∅} ∨ 𝑥𝐵)))
4233, 41ax-mp 5 . . . . . 6 (𝑥𝐵 ↔ ({∅} ∈ {∅} ∨ 𝑥𝐵))
4339, 40, 423bitr4ri 293 . . . . 5 (𝑥𝐵 ↔ ⟨𝑥, {∅}⟩ ∈ ((𝐴 × {∅}) ∪ (𝐵 × {{∅}})))
44 opelxp 5136 . . . . . . . 8 (⟨𝑥, {∅}⟩ ∈ (𝐶 × {∅}) ↔ (𝑥𝐶 ∧ {∅} ∈ {∅}))
4533bianfi 965 . . . . . . . 8 ({∅} ∈ {∅} ↔ (𝑥𝐶 ∧ {∅} ∈ {∅}))
4644, 45bitr4i 267 . . . . . . 7 (⟨𝑥, {∅}⟩ ∈ (𝐶 × {∅}) ↔ {∅} ∈ {∅})
47 opelxp 5136 . . . . . . . 8 (⟨𝑥, {∅}⟩ ∈ (𝐷 × {{∅}}) ↔ (𝑥𝐷 ∧ {∅} ∈ {{∅}}))
4836, 47mpbiran2 953 . . . . . . 7 (⟨𝑥, {∅}⟩ ∈ (𝐷 × {{∅}}) ↔ 𝑥𝐷)
4946, 48orbi12i 543 . . . . . 6 ((⟨𝑥, {∅}⟩ ∈ (𝐶 × {∅}) ∨ ⟨𝑥, {∅}⟩ ∈ (𝐷 × {{∅}})) ↔ ({∅} ∈ {∅} ∨ 𝑥𝐷))
50 elun 3745 . . . . . 6 (⟨𝑥, {∅}⟩ ∈ ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})) ↔ (⟨𝑥, {∅}⟩ ∈ (𝐶 × {∅}) ∨ ⟨𝑥, {∅}⟩ ∈ (𝐷 × {{∅}})))
51 biorf 420 . . . . . . 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 2618 . . 3 (((𝐴 × {∅}) ∪ (𝐵 × {{∅}})) = ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})) → 𝐵 = 𝐷)
5626, 55jca 554 . 2 (((𝐴 × {∅}) ∪ (𝐵 × {{∅}})) = ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})) → (𝐴 = 𝐶𝐵 = 𝐷))
57 xpeq1 5118 . . 3 (𝐴 = 𝐶 → (𝐴 × {∅}) = (𝐶 × {∅}))
58 xpeq1 5118 . . 3 (𝐵 = 𝐷 → (𝐵 × {{∅}}) = (𝐷 × {{∅}}))
59 uneq12 3754 . . 3 (((𝐴 × {∅}) = (𝐶 × {∅}) ∧ (𝐵 × {{∅}}) = (𝐷 × {{∅}})) → ((𝐴 × {∅}) ∪ (𝐵 × {{∅}})) = ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})))
6057, 58, 59syl2an 494 . 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 383  wa 384   = wceq 1481  wcel 1988  cun 3565  c0 3907  {csn 4168  cop 4174   × cxp 5102
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-opab 4704  df-xp 5110
This theorem is referenced by: (None)
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