Theorem opthprc 5696

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 2826	. . . . 5 ⊢ (((𝐴 × {∅}) ∪ (𝐵 × {{∅}})) = ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})) → (⟨𝑥, ∅⟩ ∈ ((𝐴 × {∅}) ∪ (𝐵 × {{∅}})) ↔ ⟨𝑥, ∅⟩ ∈ ((𝐶 × {∅}) ∪ (𝐷 × {{∅}}))))
2		0ex 5254	. . . . . . . . 9 ⊢ ∅ ∈ V
3	2	snid 4621	. . . . . . . 8 ⊢ ∅ ∈ {∅}
4		opelxp 5668	. . . . . . . 8 ⊢ (⟨𝑥, ∅⟩ ∈ (𝐴 × {∅}) ↔ (𝑥 ∈ 𝐴 ∧ ∅ ∈ {∅}))
5	3, 4	mpbiran2 711	. . . . . . 7 ⊢ (⟨𝑥, ∅⟩ ∈ (𝐴 × {∅}) ↔ 𝑥 ∈ 𝐴)
6		opelxp 5668	. . . . . . . 8 ⊢ (⟨𝑥, ∅⟩ ∈ (𝐵 × {{∅}}) ↔ (𝑥 ∈ 𝐵 ∧ ∅ ∈ {{∅}}))
7		0nep0 5305	. . . . . . . . . 10 ⊢ ∅ ≠ {∅}
8	2	elsn 4597	. . . . . . . . . 10 ⊢ (∅ ∈ {{∅}} ↔ ∅ = {∅})
9	7, 8	nemtbir 3029	. . . . . . . . 9 ⊢ ¬ ∅ ∈ {{∅}}
10	9	bianfi 533	. . . . . . . 8 ⊢ (∅ ∈ {{∅}} ↔ (𝑥 ∈ 𝐵 ∧ ∅ ∈ {{∅}}))
11	6, 10	bitr4i 278	. . . . . . 7 ⊢ (⟨𝑥, ∅⟩ ∈ (𝐵 × {{∅}}) ↔ ∅ ∈ {{∅}})
12	5, 11	orbi12i 915	. . . . . 6 ⊢ ((⟨𝑥, ∅⟩ ∈ (𝐴 × {∅}) ∨ ⟨𝑥, ∅⟩ ∈ (𝐵 × {{∅}})) ↔ (𝑥 ∈ 𝐴 ∨ ∅ ∈ {{∅}}))
13		elun 4107	. . . . . 6 ⊢ (⟨𝑥, ∅⟩ ∈ ((𝐴 × {∅}) ∪ (𝐵 × {{∅}})) ↔ (⟨𝑥, ∅⟩ ∈ (𝐴 × {∅}) ∨ ⟨𝑥, ∅⟩ ∈ (𝐵 × {{∅}})))
14	9	biorfri 940	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∨ ∅ ∈ {{∅}}))
15	12, 13, 14	3bitr4ri 304	. . . . 5 ⊢ (𝑥 ∈ 𝐴 ↔ ⟨𝑥, ∅⟩ ∈ ((𝐴 × {∅}) ∪ (𝐵 × {{∅}})))
16		opelxp 5668	. . . . . . . 8 ⊢ (⟨𝑥, ∅⟩ ∈ (𝐶 × {∅}) ↔ (𝑥 ∈ 𝐶 ∧ ∅ ∈ {∅}))
17	3, 16	mpbiran2 711	. . . . . . 7 ⊢ (⟨𝑥, ∅⟩ ∈ (𝐶 × {∅}) ↔ 𝑥 ∈ 𝐶)
18		opelxp 5668	. . . . . . . 8 ⊢ (⟨𝑥, ∅⟩ ∈ (𝐷 × {{∅}}) ↔ (𝑥 ∈ 𝐷 ∧ ∅ ∈ {{∅}}))
19	9	bianfi 533	. . . . . . . 8 ⊢ (∅ ∈ {{∅}} ↔ (𝑥 ∈ 𝐷 ∧ ∅ ∈ {{∅}}))
20	18, 19	bitr4i 278	. . . . . . 7 ⊢ (⟨𝑥, ∅⟩ ∈ (𝐷 × {{∅}}) ↔ ∅ ∈ {{∅}})
21	17, 20	orbi12i 915	. . . . . 6 ⊢ ((⟨𝑥, ∅⟩ ∈ (𝐶 × {∅}) ∨ ⟨𝑥, ∅⟩ ∈ (𝐷 × {{∅}})) ↔ (𝑥 ∈ 𝐶 ∨ ∅ ∈ {{∅}}))
22		elun 4107	. . . . . 6 ⊢ (⟨𝑥, ∅⟩ ∈ ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})) ↔ (⟨𝑥, ∅⟩ ∈ (𝐶 × {∅}) ∨ ⟨𝑥, ∅⟩ ∈ (𝐷 × {{∅}})))
23	9	biorfri 940	. . . . . 6 ⊢ (𝑥 ∈ 𝐶 ↔ (𝑥 ∈ 𝐶 ∨ ∅ ∈ {{∅}}))
24	21, 22, 23	3bitr4ri 304	. . . . 5 ⊢ (𝑥 ∈ 𝐶 ↔ ⟨𝑥, ∅⟩ ∈ ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})))
25	1, 15, 24	3bitr4g 314	. . . 4 ⊢ (((𝐴 × {∅}) ∪ (𝐵 × {{∅}})) = ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})) → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐶))
26	25	eqrdv 2735	. . 3 ⊢ (((𝐴 × {∅}) ∪ (𝐵 × {{∅}})) = ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})) → 𝐴 = 𝐶)
27		eleq2 2826	. . . . 5 ⊢ (((𝐴 × {∅}) ∪ (𝐵 × {{∅}})) = ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})) → (⟨𝑥, {∅}⟩ ∈ ((𝐴 × {∅}) ∪ (𝐵 × {{∅}})) ↔ ⟨𝑥, {∅}⟩ ∈ ((𝐶 × {∅}) ∪ (𝐷 × {{∅}}))))
28		opelxp 5668	. . . . . . . 8 ⊢ (⟨𝑥, {∅}⟩ ∈ (𝐴 × {∅}) ↔ (𝑥 ∈ 𝐴 ∧ {∅} ∈ {∅}))
29		snex 5385	. . . . . . . . . . . 12 ⊢ {∅} ∈ V
30	29	elsn 4597	. . . . . . . . . . 11 ⊢ ({∅} ∈ {∅} ↔ {∅} = ∅)
31		eqcom 2744	. . . . . . . . . . 11 ⊢ ({∅} = ∅ ↔ ∅ = {∅})
32	30, 31	bitri 275	. . . . . . . . . 10 ⊢ ({∅} ∈ {∅} ↔ ∅ = {∅})
33	7, 32	nemtbir 3029	. . . . . . . . 9 ⊢ ¬ {∅} ∈ {∅}
34	33	bianfi 533	. . . . . . . 8 ⊢ ({∅} ∈ {∅} ↔ (𝑥 ∈ 𝐴 ∧ {∅} ∈ {∅}))
35	28, 34	bitr4i 278	. . . . . . 7 ⊢ (⟨𝑥, {∅}⟩ ∈ (𝐴 × {∅}) ↔ {∅} ∈ {∅})
36	29	snid 4621	. . . . . . . 8 ⊢ {∅} ∈ {{∅}}
37		opelxp 5668	. . . . . . . 8 ⊢ (⟨𝑥, {∅}⟩ ∈ (𝐵 × {{∅}}) ↔ (𝑥 ∈ 𝐵 ∧ {∅} ∈ {{∅}}))
38	36, 37	mpbiran2 711	. . . . . . 7 ⊢ (⟨𝑥, {∅}⟩ ∈ (𝐵 × {{∅}}) ↔ 𝑥 ∈ 𝐵)
39	35, 38	orbi12i 915	. . . . . 6 ⊢ ((⟨𝑥, {∅}⟩ ∈ (𝐴 × {∅}) ∨ ⟨𝑥, {∅}⟩ ∈ (𝐵 × {{∅}})) ↔ ({∅} ∈ {∅} ∨ 𝑥 ∈ 𝐵))
40		elun 4107	. . . . . 6 ⊢ (⟨𝑥, {∅}⟩ ∈ ((𝐴 × {∅}) ∪ (𝐵 × {{∅}})) ↔ (⟨𝑥, {∅}⟩ ∈ (𝐴 × {∅}) ∨ ⟨𝑥, {∅}⟩ ∈ (𝐵 × {{∅}})))
41	33	biorfi 939	. . . . . 6 ⊢ (𝑥 ∈ 𝐵 ↔ ({∅} ∈ {∅} ∨ 𝑥 ∈ 𝐵))
42	39, 40, 41	3bitr4ri 304	. . . . 5 ⊢ (𝑥 ∈ 𝐵 ↔ ⟨𝑥, {∅}⟩ ∈ ((𝐴 × {∅}) ∪ (𝐵 × {{∅}})))
43		opelxp 5668	. . . . . . . 8 ⊢ (⟨𝑥, {∅}⟩ ∈ (𝐶 × {∅}) ↔ (𝑥 ∈ 𝐶 ∧ {∅} ∈ {∅}))
44	33	bianfi 533	. . . . . . . 8 ⊢ ({∅} ∈ {∅} ↔ (𝑥 ∈ 𝐶 ∧ {∅} ∈ {∅}))
45	43, 44	bitr4i 278	. . . . . . 7 ⊢ (⟨𝑥, {∅}⟩ ∈ (𝐶 × {∅}) ↔ {∅} ∈ {∅})
46		opelxp 5668	. . . . . . . 8 ⊢ (⟨𝑥, {∅}⟩ ∈ (𝐷 × {{∅}}) ↔ (𝑥 ∈ 𝐷 ∧ {∅} ∈ {{∅}}))
47	36, 46	mpbiran2 711	. . . . . . 7 ⊢ (⟨𝑥, {∅}⟩ ∈ (𝐷 × {{∅}}) ↔ 𝑥 ∈ 𝐷)
48	45, 47	orbi12i 915	. . . . . 6 ⊢ ((⟨𝑥, {∅}⟩ ∈ (𝐶 × {∅}) ∨ ⟨𝑥, {∅}⟩ ∈ (𝐷 × {{∅}})) ↔ ({∅} ∈ {∅} ∨ 𝑥 ∈ 𝐷))
49		elun 4107	. . . . . 6 ⊢ (⟨𝑥, {∅}⟩ ∈ ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})) ↔ (⟨𝑥, {∅}⟩ ∈ (𝐶 × {∅}) ∨ ⟨𝑥, {∅}⟩ ∈ (𝐷 × {{∅}})))
50	33	biorfi 939	. . . . . 6 ⊢ (𝑥 ∈ 𝐷 ↔ ({∅} ∈ {∅} ∨ 𝑥 ∈ 𝐷))
51	48, 49, 50	3bitr4ri 304	. . . . 5 ⊢ (𝑥 ∈ 𝐷 ↔ ⟨𝑥, {∅}⟩ ∈ ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})))
52	27, 42, 51	3bitr4g 314	. . . 4 ⊢ (((𝐴 × {∅}) ∪ (𝐵 × {{∅}})) = ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})) → (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ 𝐷))
53	52	eqrdv 2735	. . 3 ⊢ (((𝐴 × {∅}) ∪ (𝐵 × {{∅}})) = ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})) → 𝐵 = 𝐷)
54	26, 53	jca 511	. 2 ⊢ (((𝐴 × {∅}) ∪ (𝐵 × {{∅}})) = ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))
55		xpeq1 5646	. . 3 ⊢ (𝐴 = 𝐶 → (𝐴 × {∅}) = (𝐶 × {∅}))
56		xpeq1 5646	. . 3 ⊢ (𝐵 = 𝐷 → (𝐵 × {{∅}}) = (𝐷 × {{∅}}))
57		uneq12 4117	. . 3 ⊢ (((𝐴 × {∅}) = (𝐶 × {∅}) ∧ (𝐵 × {{∅}}) = (𝐷 × {{∅}})) → ((𝐴 × {∅}) ∪ (𝐵 × {{∅}})) = ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})))
58	55, 56, 57	syl2an 597	. 2 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → ((𝐴 × {∅}) ∪ (𝐵 × {{∅}})) = ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})))
59	54, 58	impbii 209	1 ⊢ (((𝐴 × {∅}) ∪ (𝐵 × {{∅}})) = ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∨ wo 848   = wceq 1542   ∈ wcel 2114   ∪ cun 3901  ∅c0 4287  {csn 4582  ⟨cop 4588   × cxp 5630

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-opab 5163  df-xp 5638

This theorem is referenced by: (None)