Theorem opthprc 5714

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.

Step	Hyp	Ref	Expression
1		eleq2 2849	. . . . 5 ⊢ (((𝐴 × {∅}) ∪ (𝐵 × {{∅}})) = ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})) → (⟨𝑥, ∅⟩ ∈ ((𝐴 × {∅}) ∪ (𝐵 × {{∅}})) ↔ ⟨𝑥, ∅⟩ ∈ ((𝐶 × {∅}) ∪ (𝐷 × {{∅}}))))
2		0ex 5260	. . . . . . . . 9 ⊢ ∅ ∈ V
3	2	snid 4619	. . . . . . . 8 ⊢ ∅ ∈ {∅}
4		opelxp 5686	. . . . . . . 8 ⊢ (⟨𝑥, ∅⟩ ∈ (𝐴 × {∅}) ↔ (𝑥 ∈ 𝐴 ∧ ∅ ∈ {∅}))
5	3, 4	mpbiran2 723	. . . . . . 7 ⊢ (⟨𝑥, ∅⟩ ∈ (𝐴 × {∅}) ↔ 𝑥 ∈ 𝐴)
6		opelxp 5686	. . . . . . . 8 ⊢ (⟨𝑥, ∅⟩ ∈ (𝐵 × {{∅}}) ↔ (𝑥 ∈ 𝐵 ∧ ∅ ∈ {{∅}}))
7		0nep0 5318	. . . . . . . . . 10 ⊢ ∅ ≠ {∅}
8	2	elsn 4595	. . . . . . . . . 10 ⊢ (∅ ∈ {{∅}} ↔ ∅ = {∅})
9	7, 8	nemtbir 3051	. . . . . . . . 9 ⊢ ¬ ∅ ∈ {{∅}}
10	9	bianfi 543	. . . . . . . 8 ⊢ (∅ ∈ {{∅}} ↔ (𝑥 ∈ 𝐵 ∧ ∅ ∈ {{∅}}))
11	6, 10	bitr4i 281	. . . . . . 7 ⊢ (⟨𝑥, ∅⟩ ∈ (𝐵 × {{∅}}) ↔ ∅ ∈ {{∅}})
12	5, 11	orbi12i 928	. . . . . 6 ⊢ ((⟨𝑥, ∅⟩ ∈ (𝐴 × {∅}) ∨ ⟨𝑥, ∅⟩ ∈ (𝐵 × {{∅}})) ↔ (𝑥 ∈ 𝐴 ∨ ∅ ∈ {{∅}}))
13		elun 4152	. . . . . 6 ⊢ (⟨𝑥, ∅⟩ ∈ ((𝐴 × {∅}) ∪ (𝐵 × {{∅}})) ↔ (⟨𝑥, ∅⟩ ∈ (𝐴 × {∅}) ∨ ⟨𝑥, ∅⟩ ∈ (𝐵 × {{∅}})))
14	9	biorfri 953	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∨ ∅ ∈ {{∅}}))
15	12, 13, 14	3bitr4ri 307	. . . . 5 ⊢ (𝑥 ∈ 𝐴 ↔ ⟨𝑥, ∅⟩ ∈ ((𝐴 × {∅}) ∪ (𝐵 × {{∅}})))
16		opelxp 5686	. . . . . . . 8 ⊢ (⟨𝑥, ∅⟩ ∈ (𝐶 × {∅}) ↔ (𝑥 ∈ 𝐶 ∧ ∅ ∈ {∅}))
17	3, 16	mpbiran2 723	. . . . . . 7 ⊢ (⟨𝑥, ∅⟩ ∈ (𝐶 × {∅}) ↔ 𝑥 ∈ 𝐶)
18		opelxp 5686	. . . . . . . 8 ⊢ (⟨𝑥, ∅⟩ ∈ (𝐷 × {{∅}}) ↔ (𝑥 ∈ 𝐷 ∧ ∅ ∈ {{∅}}))
19	9	bianfi 543	. . . . . . . 8 ⊢ (∅ ∈ {{∅}} ↔ (𝑥 ∈ 𝐷 ∧ ∅ ∈ {{∅}}))
20	18, 19	bitr4i 281	. . . . . . 7 ⊢ (⟨𝑥, ∅⟩ ∈ (𝐷 × {{∅}}) ↔ ∅ ∈ {{∅}})
21	17, 20	orbi12i 928	. . . . . 6 ⊢ ((⟨𝑥, ∅⟩ ∈ (𝐶 × {∅}) ∨ ⟨𝑥, ∅⟩ ∈ (𝐷 × {{∅}})) ↔ (𝑥 ∈ 𝐶 ∨ ∅ ∈ {{∅}}))
22		elun 4152	. . . . . 6 ⊢ (⟨𝑥, ∅⟩ ∈ ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})) ↔ (⟨𝑥, ∅⟩ ∈ (𝐶 × {∅}) ∨ ⟨𝑥, ∅⟩ ∈ (𝐷 × {{∅}})))
23	9	biorfri 953	. . . . . 6 ⊢ (𝑥 ∈ 𝐶 ↔ (𝑥 ∈ 𝐶 ∨ ∅ ∈ {{∅}}))
24	21, 22, 23	3bitr4ri 307	. . . . 5 ⊢ (𝑥 ∈ 𝐶 ↔ ⟨𝑥, ∅⟩ ∈ ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})))
25	1, 15, 24	3bitr4g 317	. . . 4 ⊢ (((𝐴 × {∅}) ∪ (𝐵 × {{∅}})) = ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})) → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐶))
26	25	eqrdv 2758	. . 3 ⊢ (((𝐴 × {∅}) ∪ (𝐵 × {{∅}})) = ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})) → 𝐴 = 𝐶)
27		eleq2 2849	. . . . 5 ⊢ (((𝐴 × {∅}) ∪ (𝐵 × {{∅}})) = ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})) → (⟨𝑥, {∅}⟩ ∈ ((𝐴 × {∅}) ∪ (𝐵 × {{∅}})) ↔ ⟨𝑥, {∅}⟩ ∈ ((𝐶 × {∅}) ∪ (𝐷 × {{∅}}))))
28		opelxp 5686	. . . . . . . 8 ⊢ (⟨𝑥, {∅}⟩ ∈ (𝐴 × {∅}) ↔ (𝑥 ∈ 𝐴 ∧ {∅} ∈ {∅}))
29		snex 5396	. . . . . . . . . . . 12 ⊢ {∅} ∈ V
30	29	elsn 4595	. . . . . . . . . . 11 ⊢ ({∅} ∈ {∅} ↔ {∅} = ∅)
31		eqcom 2767	. . . . . . . . . . 11 ⊢ ({∅} = ∅ ↔ ∅ = {∅})
32	30, 31	bitri 278	. . . . . . . . . 10 ⊢ ({∅} ∈ {∅} ↔ ∅ = {∅})
33	7, 32	nemtbir 3051	. . . . . . . . 9 ⊢ ¬ {∅} ∈ {∅}
34	33	bianfi 543	. . . . . . . 8 ⊢ ({∅} ∈ {∅} ↔ (𝑥 ∈ 𝐴 ∧ {∅} ∈ {∅}))
35	28, 34	bitr4i 281	. . . . . . 7 ⊢ (⟨𝑥, {∅}⟩ ∈ (𝐴 × {∅}) ↔ {∅} ∈ {∅})
36	29	snid 4619	. . . . . . . 8 ⊢ {∅} ∈ {{∅}}
37		opelxp 5686	. . . . . . . 8 ⊢ (⟨𝑥, {∅}⟩ ∈ (𝐵 × {{∅}}) ↔ (𝑥 ∈ 𝐵 ∧ {∅} ∈ {{∅}}))
38	36, 37	mpbiran2 723	. . . . . . 7 ⊢ (⟨𝑥, {∅}⟩ ∈ (𝐵 × {{∅}}) ↔ 𝑥 ∈ 𝐵)
39	35, 38	orbi12i 928	. . . . . 6 ⊢ ((⟨𝑥, {∅}⟩ ∈ (𝐴 × {∅}) ∨ ⟨𝑥, {∅}⟩ ∈ (𝐵 × {{∅}})) ↔ ({∅} ∈ {∅} ∨ 𝑥 ∈ 𝐵))
40		elun 4152	. . . . . 6 ⊢ (⟨𝑥, {∅}⟩ ∈ ((𝐴 × {∅}) ∪ (𝐵 × {{∅}})) ↔ (⟨𝑥, {∅}⟩ ∈ (𝐴 × {∅}) ∨ ⟨𝑥, {∅}⟩ ∈ (𝐵 × {{∅}})))
41	33	biorfi 952	. . . . . 6 ⊢ (𝑥 ∈ 𝐵 ↔ ({∅} ∈ {∅} ∨ 𝑥 ∈ 𝐵))
42	39, 40, 41	3bitr4ri 307	. . . . 5 ⊢ (𝑥 ∈ 𝐵 ↔ ⟨𝑥, {∅}⟩ ∈ ((𝐴 × {∅}) ∪ (𝐵 × {{∅}})))
43		opelxp 5686	. . . . . . . 8 ⊢ (⟨𝑥, {∅}⟩ ∈ (𝐶 × {∅}) ↔ (𝑥 ∈ 𝐶 ∧ {∅} ∈ {∅}))
44	33	bianfi 543	. . . . . . . 8 ⊢ ({∅} ∈ {∅} ↔ (𝑥 ∈ 𝐶 ∧ {∅} ∈ {∅}))
45	43, 44	bitr4i 281	. . . . . . 7 ⊢ (⟨𝑥, {∅}⟩ ∈ (𝐶 × {∅}) ↔ {∅} ∈ {∅})
46		opelxp 5686	. . . . . . . 8 ⊢ (⟨𝑥, {∅}⟩ ∈ (𝐷 × {{∅}}) ↔ (𝑥 ∈ 𝐷 ∧ {∅} ∈ {{∅}}))
47	36, 46	mpbiran2 723	. . . . . . 7 ⊢ (⟨𝑥, {∅}⟩ ∈ (𝐷 × {{∅}}) ↔ 𝑥 ∈ 𝐷)
48	45, 47	orbi12i 928	. . . . . 6 ⊢ ((⟨𝑥, {∅}⟩ ∈ (𝐶 × {∅}) ∨ ⟨𝑥, {∅}⟩ ∈ (𝐷 × {{∅}})) ↔ ({∅} ∈ {∅} ∨ 𝑥 ∈ 𝐷))
49		elun 4152	. . . . . 6 ⊢ (⟨𝑥, {∅}⟩ ∈ ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})) ↔ (⟨𝑥, {∅}⟩ ∈ (𝐶 × {∅}) ∨ ⟨𝑥, {∅}⟩ ∈ (𝐷 × {{∅}})))
50	33	biorfi 952	. . . . . 6 ⊢ (𝑥 ∈ 𝐷 ↔ ({∅} ∈ {∅} ∨ 𝑥 ∈ 𝐷))
51	48, 49, 50	3bitr4ri 307	. . . . 5 ⊢ (𝑥 ∈ 𝐷 ↔ ⟨𝑥, {∅}⟩ ∈ ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})))
52	27, 42, 51	3bitr4g 317	. . . 4 ⊢ (((𝐴 × {∅}) ∪ (𝐵 × {{∅}})) = ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})) → (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ 𝐷))
53	52	eqrdv 2758	. . 3 ⊢ (((𝐴 × {∅}) ∪ (𝐵 × {{∅}})) = ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})) → 𝐵 = 𝐷)
54	26, 53	jca 521	. 2 ⊢ (((𝐴 × {∅}) ∪ (𝐵 × {{∅}})) = ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))
55		xpeq1 5664	. . 3 ⊢ (𝐴 = 𝐶 → (𝐴 × {∅}) = (𝐶 × {∅}))
56		xpeq1 5664	. . 3 ⊢ (𝐵 = 𝐷 → (𝐵 × {{∅}}) = (𝐷 × {{∅}}))
57		uneq12 4162	. . 3 ⊢ (((𝐴 × {∅}) = (𝐶 × {∅}) ∧ (𝐵 × {{∅}}) = (𝐷 × {{∅}})) → ((𝐴 × {∅}) ∪ (𝐵 × {{∅}})) = ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})))
58	55, 56, 57	syl2an 608	. 2 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → ((𝐴 × {∅}) ∪ (𝐵 × {{∅}})) = ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})))
59	54, 58	impbii 212	1 ⊢ (((𝐴 × {∅}) ∪ (𝐵 × {{∅}})) = ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1569   ∈ wcel 2144   ∪ cun 4149  ∅c0 4278  {csn 4584  ⟨cop 4834   × cxp 5648

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-opab 5167  df-xp 5656

This theorem is used by: (None)