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Theorem opthprc 4318
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 = 𝐶 B = 𝐷))

Proof of Theorem opthprc
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 eleq2 2083 . . . . 5 (((A × {∅}) ∪ (B × {{∅}})) = ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})) → (⟨x, ∅⟩ ((A × {∅}) ∪ (B × {{∅}})) ↔ ⟨x, ∅⟩ ((𝐶 × {∅}) ∪ (𝐷 × {{∅}}))))
2 0ex 3858 . . . . . . . . 9 V
32snid 3377 . . . . . . . 8 {∅}
4 opelxp 4301 . . . . . . . 8 (⟨x, ∅⟩ (A × {∅}) ↔ (x A {∅}))
53, 4mpbiran2 836 . . . . . . 7 (⟨x, ∅⟩ (A × {∅}) ↔ x A)
6 opelxp 4301 . . . . . . . 8 (⟨x, ∅⟩ (B × {{∅}}) ↔ (x B {{∅}}))
7 0nep0 3892 . . . . . . . . . 10 ∅ ≠ {∅}
82elsnc 3373 . . . . . . . . . 10 (∅ {{∅}} ↔ ∅ = {∅})
97, 8nemtbir 2272 . . . . . . . . 9 ¬ ∅ {{∅}}
109bianfi 842 . . . . . . . 8 (∅ {{∅}} ↔ (x B {{∅}}))
116, 10bitr4i 176 . . . . . . 7 (⟨x, ∅⟩ (B × {{∅}}) ↔ ∅ {{∅}})
125, 11orbi12i 668 . . . . . 6 ((⟨x, ∅⟩ (A × {∅}) x, ∅⟩ (B × {{∅}})) ↔ (x A {{∅}}))
13 elun 3061 . . . . . 6 (⟨x, ∅⟩ ((A × {∅}) ∪ (B × {{∅}})) ↔ (⟨x, ∅⟩ (A × {∅}) x, ∅⟩ (B × {{∅}})))
149biorfi 652 . . . . . 6 (x A ↔ (x A {{∅}}))
1512, 13, 143bitr4ri 202 . . . . 5 (x A ↔ ⟨x, ∅⟩ ((A × {∅}) ∪ (B × {{∅}})))
16 opelxp 4301 . . . . . . . 8 (⟨x, ∅⟩ (𝐶 × {∅}) ↔ (x 𝐶 {∅}))
173, 16mpbiran2 836 . . . . . . 7 (⟨x, ∅⟩ (𝐶 × {∅}) ↔ x 𝐶)
18 opelxp 4301 . . . . . . . 8 (⟨x, ∅⟩ (𝐷 × {{∅}}) ↔ (x 𝐷 {{∅}}))
199bianfi 842 . . . . . . . 8 (∅ {{∅}} ↔ (x 𝐷 {{∅}}))
2018, 19bitr4i 176 . . . . . . 7 (⟨x, ∅⟩ (𝐷 × {{∅}}) ↔ ∅ {{∅}})
2117, 20orbi12i 668 . . . . . 6 ((⟨x, ∅⟩ (𝐶 × {∅}) x, ∅⟩ (𝐷 × {{∅}})) ↔ (x 𝐶 {{∅}}))
22 elun 3061 . . . . . 6 (⟨x, ∅⟩ ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})) ↔ (⟨x, ∅⟩ (𝐶 × {∅}) x, ∅⟩ (𝐷 × {{∅}})))
239biorfi 652 . . . . . 6 (x 𝐶 ↔ (x 𝐶 {{∅}}))
2421, 22, 233bitr4ri 202 . . . . 5 (x 𝐶 ↔ ⟨x, ∅⟩ ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})))
251, 15, 243bitr4g 212 . . . 4 (((A × {∅}) ∪ (B × {{∅}})) = ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})) → (x Ax 𝐶))
2625eqrdv 2020 . . 3 (((A × {∅}) ∪ (B × {{∅}})) = ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})) → A = 𝐶)
27 eleq2 2083 . . . . 5 (((A × {∅}) ∪ (B × {{∅}})) = ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})) → (⟨x, {∅}⟩ ((A × {∅}) ∪ (B × {{∅}})) ↔ ⟨x, {∅}⟩ ((𝐶 × {∅}) ∪ (𝐷 × {{∅}}))))
28 opelxp 4301 . . . . . . . 8 (⟨x, {∅}⟩ (A × {∅}) ↔ (x A {∅} {∅}))
29 p0ex 3913 . . . . . . . . . . . 12 {∅} V
3029elsnc 3373 . . . . . . . . . . 11 ({∅} {∅} ↔ {∅} = ∅)
31 eqcom 2024 . . . . . . . . . . 11 ({∅} = ∅ ↔ ∅ = {∅})
3230, 31bitri 173 . . . . . . . . . 10 ({∅} {∅} ↔ ∅ = {∅})
337, 32nemtbir 2272 . . . . . . . . 9 ¬ {∅} {∅}
3433bianfi 842 . . . . . . . 8 ({∅} {∅} ↔ (x A {∅} {∅}))
3528, 34bitr4i 176 . . . . . . 7 (⟨x, {∅}⟩ (A × {∅}) ↔ {∅} {∅})
3629snid 3377 . . . . . . . 8 {∅} {{∅}}
37 opelxp 4301 . . . . . . . 8 (⟨x, {∅}⟩ (B × {{∅}}) ↔ (x B {∅} {{∅}}))
3836, 37mpbiran2 836 . . . . . . 7 (⟨x, {∅}⟩ (B × {{∅}}) ↔ x B)
3935, 38orbi12i 668 . . . . . 6 ((⟨x, {∅}⟩ (A × {∅}) x, {∅}⟩ (B × {{∅}})) ↔ ({∅} {∅} x B))
40 elun 3061 . . . . . 6 (⟨x, {∅}⟩ ((A × {∅}) ∪ (B × {{∅}})) ↔ (⟨x, {∅}⟩ (A × {∅}) x, {∅}⟩ (B × {{∅}})))
41 biorf 650 . . . . . . 7 (¬ {∅} {∅} → (x B ↔ ({∅} {∅} x B)))
4233, 41ax-mp 7 . . . . . 6 (x B ↔ ({∅} {∅} x B))
4339, 40, 423bitr4ri 202 . . . . 5 (x B ↔ ⟨x, {∅}⟩ ((A × {∅}) ∪ (B × {{∅}})))
44 opelxp 4301 . . . . . . . 8 (⟨x, {∅}⟩ (𝐶 × {∅}) ↔ (x 𝐶 {∅} {∅}))
4533bianfi 842 . . . . . . . 8 ({∅} {∅} ↔ (x 𝐶 {∅} {∅}))
4644, 45bitr4i 176 . . . . . . 7 (⟨x, {∅}⟩ (𝐶 × {∅}) ↔ {∅} {∅})
47 opelxp 4301 . . . . . . . 8 (⟨x, {∅}⟩ (𝐷 × {{∅}}) ↔ (x 𝐷 {∅} {{∅}}))
4836, 47mpbiran2 836 . . . . . . 7 (⟨x, {∅}⟩ (𝐷 × {{∅}}) ↔ x 𝐷)
4946, 48orbi12i 668 . . . . . 6 ((⟨x, {∅}⟩ (𝐶 × {∅}) x, {∅}⟩ (𝐷 × {{∅}})) ↔ ({∅} {∅} x 𝐷))
50 elun 3061 . . . . . 6 (⟨x, {∅}⟩ ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})) ↔ (⟨x, {∅}⟩ (𝐶 × {∅}) x, {∅}⟩ (𝐷 × {{∅}})))
51 biorf 650 . . . . . . 7 (¬ {∅} {∅} → (x 𝐷 ↔ ({∅} {∅} x 𝐷)))
5233, 51ax-mp 7 . . . . . 6 (x 𝐷 ↔ ({∅} {∅} x 𝐷))
5349, 50, 523bitr4ri 202 . . . . 5 (x 𝐷 ↔ ⟨x, {∅}⟩ ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})))
5427, 43, 533bitr4g 212 . . . 4 (((A × {∅}) ∪ (B × {{∅}})) = ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})) → (x Bx 𝐷))
5554eqrdv 2020 . . 3 (((A × {∅}) ∪ (B × {{∅}})) = ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})) → B = 𝐷)
5626, 55jca 290 . 2 (((A × {∅}) ∪ (B × {{∅}})) = ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})) → (A = 𝐶 B = 𝐷))
57 xpeq1 4286 . . 3 (A = 𝐶 → (A × {∅}) = (𝐶 × {∅}))
58 xpeq1 4286 . . 3 (B = 𝐷 → (B × {{∅}}) = (𝐷 × {{∅}}))
59 uneq12 3069 . . 3 (((A × {∅}) = (𝐶 × {∅}) (B × {{∅}}) = (𝐷 × {{∅}})) → ((A × {∅}) ∪ (B × {{∅}})) = ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})))
6057, 58, 59syl2an 273 . 2 ((A = 𝐶 B = 𝐷) → ((A × {∅}) ∪ (B × {{∅}})) = ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})))
6156, 60impbii 117 1 (((A × {∅}) ∪ (B × {{∅}})) = ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})) ↔ (A = 𝐶 B = 𝐷))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3   wa 97  wb 98   wo 616   = wceq 1228   wcel 1374  cun 2892  c0 3201  {csn 3350  cop 3353   × cxp 4270
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-nul 3857  ax-pow 3901  ax-pr 3918
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ne 2188  df-ral 2289  df-rex 2290  df-v 2537  df-dif 2897  df-un 2899  df-in 2901  df-ss 2908  df-nul 3202  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-opab 3793  df-xp 4278
This theorem is referenced by: (None)
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