Theorem opthprc 4585

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 2201	. . . . 5 ⊢ (((𝐴 × {∅}) ∪ (𝐵 × {{∅}})) = ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})) → (⟨𝑥, ∅⟩ ∈ ((𝐴 × {∅}) ∪ (𝐵 × {{∅}})) ↔ ⟨𝑥, ∅⟩ ∈ ((𝐶 × {∅}) ∪ (𝐷 × {{∅}}))))
2		0ex 4050	. . . . . . . . 9 ⊢ ∅ ∈ V
3	2	snid 3551	. . . . . . . 8 ⊢ ∅ ∈ {∅}
4		opelxp 4564	. . . . . . . 8 ⊢ (⟨𝑥, ∅⟩ ∈ (𝐴 × {∅}) ↔ (𝑥 ∈ 𝐴 ∧ ∅ ∈ {∅}))
5	3, 4	mpbiran2 925	. . . . . . 7 ⊢ (⟨𝑥, ∅⟩ ∈ (𝐴 × {∅}) ↔ 𝑥 ∈ 𝐴)
6		opelxp 4564	. . . . . . . 8 ⊢ (⟨𝑥, ∅⟩ ∈ (𝐵 × {{∅}}) ↔ (𝑥 ∈ 𝐵 ∧ ∅ ∈ {{∅}}))
7		0nep0 4084	. . . . . . . . . 10 ⊢ ∅ ≠ {∅}
8	2	elsn 3538	. . . . . . . . . 10 ⊢ (∅ ∈ {{∅}} ↔ ∅ = {∅})
9	7, 8	nemtbir 2395	. . . . . . . . 9 ⊢ ¬ ∅ ∈ {{∅}}
10	9	bianfi 931	. . . . . . . 8 ⊢ (∅ ∈ {{∅}} ↔ (𝑥 ∈ 𝐵 ∧ ∅ ∈ {{∅}}))
11	6, 10	bitr4i 186	. . . . . . 7 ⊢ (⟨𝑥, ∅⟩ ∈ (𝐵 × {{∅}}) ↔ ∅ ∈ {{∅}})
12	5, 11	orbi12i 753	. . . . . 6 ⊢ ((⟨𝑥, ∅⟩ ∈ (𝐴 × {∅}) ∨ ⟨𝑥, ∅⟩ ∈ (𝐵 × {{∅}})) ↔ (𝑥 ∈ 𝐴 ∨ ∅ ∈ {{∅}}))
13		elun 3212	. . . . . 6 ⊢ (⟨𝑥, ∅⟩ ∈ ((𝐴 × {∅}) ∪ (𝐵 × {{∅}})) ↔ (⟨𝑥, ∅⟩ ∈ (𝐴 × {∅}) ∨ ⟨𝑥, ∅⟩ ∈ (𝐵 × {{∅}})))
14	9	biorfi 735	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∨ ∅ ∈ {{∅}}))
15	12, 13, 14	3bitr4ri 212	. . . . 5 ⊢ (𝑥 ∈ 𝐴 ↔ ⟨𝑥, ∅⟩ ∈ ((𝐴 × {∅}) ∪ (𝐵 × {{∅}})))
16		opelxp 4564	. . . . . . . 8 ⊢ (⟨𝑥, ∅⟩ ∈ (𝐶 × {∅}) ↔ (𝑥 ∈ 𝐶 ∧ ∅ ∈ {∅}))
17	3, 16	mpbiran2 925	. . . . . . 7 ⊢ (⟨𝑥, ∅⟩ ∈ (𝐶 × {∅}) ↔ 𝑥 ∈ 𝐶)
18		opelxp 4564	. . . . . . . 8 ⊢ (⟨𝑥, ∅⟩ ∈ (𝐷 × {{∅}}) ↔ (𝑥 ∈ 𝐷 ∧ ∅ ∈ {{∅}}))
19	9	bianfi 931	. . . . . . . 8 ⊢ (∅ ∈ {{∅}} ↔ (𝑥 ∈ 𝐷 ∧ ∅ ∈ {{∅}}))
20	18, 19	bitr4i 186	. . . . . . 7 ⊢ (⟨𝑥, ∅⟩ ∈ (𝐷 × {{∅}}) ↔ ∅ ∈ {{∅}})
21	17, 20	orbi12i 753	. . . . . 6 ⊢ ((⟨𝑥, ∅⟩ ∈ (𝐶 × {∅}) ∨ ⟨𝑥, ∅⟩ ∈ (𝐷 × {{∅}})) ↔ (𝑥 ∈ 𝐶 ∨ ∅ ∈ {{∅}}))
22		elun 3212	. . . . . 6 ⊢ (⟨𝑥, ∅⟩ ∈ ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})) ↔ (⟨𝑥, ∅⟩ ∈ (𝐶 × {∅}) ∨ ⟨𝑥, ∅⟩ ∈ (𝐷 × {{∅}})))
23	9	biorfi 735	. . . . . 6 ⊢ (𝑥 ∈ 𝐶 ↔ (𝑥 ∈ 𝐶 ∨ ∅ ∈ {{∅}}))
24	21, 22, 23	3bitr4ri 212	. . . . 5 ⊢ (𝑥 ∈ 𝐶 ↔ ⟨𝑥, ∅⟩ ∈ ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})))
25	1, 15, 24	3bitr4g 222	. . . 4 ⊢ (((𝐴 × {∅}) ∪ (𝐵 × {{∅}})) = ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})) → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐶))
26	25	eqrdv 2135	. . 3 ⊢ (((𝐴 × {∅}) ∪ (𝐵 × {{∅}})) = ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})) → 𝐴 = 𝐶)
27		eleq2 2201	. . . . 5 ⊢ (((𝐴 × {∅}) ∪ (𝐵 × {{∅}})) = ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})) → (⟨𝑥, {∅}⟩ ∈ ((𝐴 × {∅}) ∪ (𝐵 × {{∅}})) ↔ ⟨𝑥, {∅}⟩ ∈ ((𝐶 × {∅}) ∪ (𝐷 × {{∅}}))))
28		opelxp 4564	. . . . . . . 8 ⊢ (⟨𝑥, {∅}⟩ ∈ (𝐴 × {∅}) ↔ (𝑥 ∈ 𝐴 ∧ {∅} ∈ {∅}))
29		p0ex 4107	. . . . . . . . . . . 12 ⊢ {∅} ∈ V
30	29	elsn 3538	. . . . . . . . . . 11 ⊢ ({∅} ∈ {∅} ↔ {∅} = ∅)
31		eqcom 2139	. . . . . . . . . . 11 ⊢ ({∅} = ∅ ↔ ∅ = {∅})
32	30, 31	bitri 183	. . . . . . . . . 10 ⊢ ({∅} ∈ {∅} ↔ ∅ = {∅})
33	7, 32	nemtbir 2395	. . . . . . . . 9 ⊢ ¬ {∅} ∈ {∅}
34	33	bianfi 931	. . . . . . . 8 ⊢ ({∅} ∈ {∅} ↔ (𝑥 ∈ 𝐴 ∧ {∅} ∈ {∅}))
35	28, 34	bitr4i 186	. . . . . . 7 ⊢ (⟨𝑥, {∅}⟩ ∈ (𝐴 × {∅}) ↔ {∅} ∈ {∅})
36	29	snid 3551	. . . . . . . 8 ⊢ {∅} ∈ {{∅}}
37		opelxp 4564	. . . . . . . 8 ⊢ (⟨𝑥, {∅}⟩ ∈ (𝐵 × {{∅}}) ↔ (𝑥 ∈ 𝐵 ∧ {∅} ∈ {{∅}}))
38	36, 37	mpbiran2 925	. . . . . . 7 ⊢ (⟨𝑥, {∅}⟩ ∈ (𝐵 × {{∅}}) ↔ 𝑥 ∈ 𝐵)
39	35, 38	orbi12i 753	. . . . . 6 ⊢ ((⟨𝑥, {∅}⟩ ∈ (𝐴 × {∅}) ∨ ⟨𝑥, {∅}⟩ ∈ (𝐵 × {{∅}})) ↔ ({∅} ∈ {∅} ∨ 𝑥 ∈ 𝐵))
40		elun 3212	. . . . . 6 ⊢ (⟨𝑥, {∅}⟩ ∈ ((𝐴 × {∅}) ∪ (𝐵 × {{∅}})) ↔ (⟨𝑥, {∅}⟩ ∈ (𝐴 × {∅}) ∨ ⟨𝑥, {∅}⟩ ∈ (𝐵 × {{∅}})))
41		biorf 733	. . . . . . 7 ⊢ (¬ {∅} ∈ {∅} → (𝑥 ∈ 𝐵 ↔ ({∅} ∈ {∅} ∨ 𝑥 ∈ 𝐵)))
42	33, 41	ax-mp 5	. . . . . 6 ⊢ (𝑥 ∈ 𝐵 ↔ ({∅} ∈ {∅} ∨ 𝑥 ∈ 𝐵))
43	39, 40, 42	3bitr4ri 212	. . . . 5 ⊢ (𝑥 ∈ 𝐵 ↔ ⟨𝑥, {∅}⟩ ∈ ((𝐴 × {∅}) ∪ (𝐵 × {{∅}})))
44		opelxp 4564	. . . . . . . 8 ⊢ (⟨𝑥, {∅}⟩ ∈ (𝐶 × {∅}) ↔ (𝑥 ∈ 𝐶 ∧ {∅} ∈ {∅}))
45	33	bianfi 931	. . . . . . . 8 ⊢ ({∅} ∈ {∅} ↔ (𝑥 ∈ 𝐶 ∧ {∅} ∈ {∅}))
46	44, 45	bitr4i 186	. . . . . . 7 ⊢ (⟨𝑥, {∅}⟩ ∈ (𝐶 × {∅}) ↔ {∅} ∈ {∅})
47		opelxp 4564	. . . . . . . 8 ⊢ (⟨𝑥, {∅}⟩ ∈ (𝐷 × {{∅}}) ↔ (𝑥 ∈ 𝐷 ∧ {∅} ∈ {{∅}}))
48	36, 47	mpbiran2 925	. . . . . . 7 ⊢ (⟨𝑥, {∅}⟩ ∈ (𝐷 × {{∅}}) ↔ 𝑥 ∈ 𝐷)
49	46, 48	orbi12i 753	. . . . . 6 ⊢ ((⟨𝑥, {∅}⟩ ∈ (𝐶 × {∅}) ∨ ⟨𝑥, {∅}⟩ ∈ (𝐷 × {{∅}})) ↔ ({∅} ∈ {∅} ∨ 𝑥 ∈ 𝐷))
50		elun 3212	. . . . . 6 ⊢ (⟨𝑥, {∅}⟩ ∈ ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})) ↔ (⟨𝑥, {∅}⟩ ∈ (𝐶 × {∅}) ∨ ⟨𝑥, {∅}⟩ ∈ (𝐷 × {{∅}})))
51		biorf 733	. . . . . . 7 ⊢ (¬ {∅} ∈ {∅} → (𝑥 ∈ 𝐷 ↔ ({∅} ∈ {∅} ∨ 𝑥 ∈ 𝐷)))
52	33, 51	ax-mp 5	. . . . . 6 ⊢ (𝑥 ∈ 𝐷 ↔ ({∅} ∈ {∅} ∨ 𝑥 ∈ 𝐷))
53	49, 50, 52	3bitr4ri 212	. . . . 5 ⊢ (𝑥 ∈ 𝐷 ↔ ⟨𝑥, {∅}⟩ ∈ ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})))
54	27, 43, 53	3bitr4g 222	. . . 4 ⊢ (((𝐴 × {∅}) ∪ (𝐵 × {{∅}})) = ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})) → (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ 𝐷))
55	54	eqrdv 2135	. . 3 ⊢ (((𝐴 × {∅}) ∪ (𝐵 × {{∅}})) = ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})) → 𝐵 = 𝐷)
56	26, 55	jca 304	. 2 ⊢ (((𝐴 × {∅}) ∪ (𝐵 × {{∅}})) = ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))
57		xpeq1 4548	. . 3 ⊢ (𝐴 = 𝐶 → (𝐴 × {∅}) = (𝐶 × {∅}))
58		xpeq1 4548	. . 3 ⊢ (𝐵 = 𝐷 → (𝐵 × {{∅}}) = (𝐷 × {{∅}}))
59		uneq12 3220	. . 3 ⊢ (((𝐴 × {∅}) = (𝐶 × {∅}) ∧ (𝐵 × {{∅}}) = (𝐷 × {{∅}})) → ((𝐴 × {∅}) ∪ (𝐵 × {{∅}})) = ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})))
60	57, 58, 59	syl2an 287	. 2 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → ((𝐴 × {∅}) ∪ (𝐵 × {{∅}})) = ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})))
61	56, 60	impbii 125	1 ⊢ (((𝐴 × {∅}) ∪ (𝐵 × {{∅}})) = ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))

Syntax hints:  ¬ wn 3   ∧ wa 103   ↔ wb 104   ∨ wo 697   = wceq 1331   ∈ wcel 1480   ∪ cun 3064  ∅c0 3358  {csn 3522  ⟨cop 3525   × cxp 4532

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-v 2683  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-opab 3985  df-xp 4540

This theorem is referenced by: (None)