Theorem opthhausdorff 5522

Description: Justification theorem for the ordered pair definition of Felix Hausdorff in "Grundzüge der Mengenlehre" ("Basics of Set Theory"), 1914, p. 32: ⟨𝐴, 𝐵⟩_H = {{𝐴, 𝑂}, {𝐵, 𝑇}}. Hausdorff used 1 and 2 instead of 𝑂 and 𝑇, but actually, any two different fixed sets will do (e.g., 𝑂 = ∅ and 𝑇 = {∅}, see 0nep0 5358). Furthermore, Hausdorff demanded that 𝑂 and 𝑇 are both different from 𝐴 as well as 𝐵, which is actually not necessary in full extent (𝐴 ≠ 𝑇 is not required). This definition is meaningful only for classes 𝐴 and 𝐵 that exist as sets (i.e., are not proper classes): If 𝐴 and 𝐶 were different proper classes (𝐴 ≠ 𝐶), then {{𝐴, 𝑂}, {𝐵, 𝑇}} = {{𝐶, 𝑂}, {𝐷, 𝑇} ↔ {{𝑂}, {𝐵, 𝑇}} = {{𝑂}, {𝐷, 𝑇} is true if 𝐵 = 𝐷, but (𝐴 = 𝐶 ∧ 𝐵 = 𝐷) would be false. See df-op 4633 for other ordered pair definitions. (Contributed by AV, 14-Jun-2022.)

Hypotheses

Ref	Expression
opthhausdorff.a	⊢ 𝐴 ∈ V
opthhausdorff.b	⊢ 𝐵 ∈ V
opthhausdorff.o	⊢ 𝐴 ≠ 𝑂
opthhausdorff.n	⊢ 𝐵 ≠ 𝑂
opthhausdorff.t	⊢ 𝐵 ≠ 𝑇
opthhausdorff.1	⊢ 𝑂 ∈ V
opthhausdorff.2	⊢ 𝑇 ∈ V
opthhausdorff.3	⊢ 𝑂 ≠ 𝑇

Assertion

Ref	Expression
opthhausdorff	⊢ ({{𝐴, 𝑂}, {𝐵, 𝑇}} = {{𝐶, 𝑂}, {𝐷, 𝑇}} ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))

Proof of Theorem opthhausdorff

Step	Hyp	Ref	Expression
1		prex 5437	. . . 4 ⊢ {𝐴, 𝑂} ∈ V
2		prex 5437	. . . 4 ⊢ {𝐵, 𝑇} ∈ V
3		opthhausdorff.a	. . . . . . 7 ⊢ 𝐴 ∈ V
4		opthhausdorff.1	. . . . . . 7 ⊢ 𝑂 ∈ V
5	3, 4	pm3.2i 470	. . . . . 6 ⊢ (𝐴 ∈ V ∧ 𝑂 ∈ V)
6		opthhausdorff.b	. . . . . . 7 ⊢ 𝐵 ∈ V
7		opthhausdorff.2	. . . . . . 7 ⊢ 𝑇 ∈ V
8	6, 7	pm3.2i 470	. . . . . 6 ⊢ (𝐵 ∈ V ∧ 𝑇 ∈ V)
9	5, 8	pm3.2i 470	. . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝑂 ∈ V) ∧ (𝐵 ∈ V ∧ 𝑇 ∈ V))
10		opthhausdorff.n	. . . . . . . 8 ⊢ 𝐵 ≠ 𝑂
11	10	necomi 2995	. . . . . . 7 ⊢ 𝑂 ≠ 𝐵
12		opthhausdorff.3	. . . . . . 7 ⊢ 𝑂 ≠ 𝑇
13	11, 12	pm3.2i 470	. . . . . 6 ⊢ (𝑂 ≠ 𝐵 ∧ 𝑂 ≠ 𝑇)
14	13	olci 867	. . . . 5 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝑇) ∨ (𝑂 ≠ 𝐵 ∧ 𝑂 ≠ 𝑇))
15		prneimg 4854	. . . . 5 ⊢ (((𝐴 ∈ V ∧ 𝑂 ∈ V) ∧ (𝐵 ∈ V ∧ 𝑇 ∈ V)) → (((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝑇) ∨ (𝑂 ≠ 𝐵 ∧ 𝑂 ≠ 𝑇)) → {𝐴, 𝑂} ≠ {𝐵, 𝑇}))
16	9, 14, 15	mp2 9	. . . 4 ⊢ {𝐴, 𝑂} ≠ {𝐵, 𝑇}
17		preq12nebg 4863	. . . 4 ⊢ (({𝐴, 𝑂} ∈ V ∧ {𝐵, 𝑇} ∈ V ∧ {𝐴, 𝑂} ≠ {𝐵, 𝑇}) → ({{𝐴, 𝑂}, {𝐵, 𝑇}} = {{𝐶, 𝑂}, {𝐷, 𝑇}} ↔ (({𝐴, 𝑂} = {𝐶, 𝑂} ∧ {𝐵, 𝑇} = {𝐷, 𝑇}) ∨ ({𝐴, 𝑂} = {𝐷, 𝑇} ∧ {𝐵, 𝑇} = {𝐶, 𝑂}))))
18	1, 2, 16, 17	mp3an 1463	. . 3 ⊢ ({{𝐴, 𝑂}, {𝐵, 𝑇}} = {{𝐶, 𝑂}, {𝐷, 𝑇}} ↔ (({𝐴, 𝑂} = {𝐶, 𝑂} ∧ {𝐵, 𝑇} = {𝐷, 𝑇}) ∨ ({𝐴, 𝑂} = {𝐷, 𝑇} ∧ {𝐵, 𝑇} = {𝐶, 𝑂})))
19		opthhausdorff.o	. . . . . 6 ⊢ 𝐴 ≠ 𝑂
20		preq12nebg 4863	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ 𝑂 ∈ V ∧ 𝐴 ≠ 𝑂) → ({𝐴, 𝑂} = {𝐶, 𝑂} ↔ ((𝐴 = 𝐶 ∧ 𝑂 = 𝑂) ∨ (𝐴 = 𝑂 ∧ 𝑂 = 𝐶))))
21	3, 4, 19, 20	mp3an 1463	. . . . 5 ⊢ ({𝐴, 𝑂} = {𝐶, 𝑂} ↔ ((𝐴 = 𝐶 ∧ 𝑂 = 𝑂) ∨ (𝐴 = 𝑂 ∧ 𝑂 = 𝐶)))
22		opthhausdorff.t	. . . . . 6 ⊢ 𝐵 ≠ 𝑇
23		preq12nebg 4863	. . . . . 6 ⊢ ((𝐵 ∈ V ∧ 𝑇 ∈ V ∧ 𝐵 ≠ 𝑇) → ({𝐵, 𝑇} = {𝐷, 𝑇} ↔ ((𝐵 = 𝐷 ∧ 𝑇 = 𝑇) ∨ (𝐵 = 𝑇 ∧ 𝑇 = 𝐷))))
24	6, 7, 22, 23	mp3an 1463	. . . . 5 ⊢ ({𝐵, 𝑇} = {𝐷, 𝑇} ↔ ((𝐵 = 𝐷 ∧ 𝑇 = 𝑇) ∨ (𝐵 = 𝑇 ∧ 𝑇 = 𝐷)))
25		simpl 482	. . . . . . 7 ⊢ ((𝐴 = 𝐶 ∧ 𝑂 = 𝑂) → 𝐴 = 𝐶)
26		simpl 482	. . . . . . 7 ⊢ ((𝐵 = 𝐷 ∧ 𝑇 = 𝑇) → 𝐵 = 𝐷)
27	25, 26	anim12i 613	. . . . . 6 ⊢ (((𝐴 = 𝐶 ∧ 𝑂 = 𝑂) ∧ (𝐵 = 𝐷 ∧ 𝑇 = 𝑇)) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))
28		eqneqall 2951	. . . . . . . 8 ⊢ (𝐴 = 𝑂 → (𝐴 ≠ 𝑂 → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))
29	19, 28	mpi 20	. . . . . . 7 ⊢ (𝐴 = 𝑂 → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))
30	29	adantr 480	. . . . . 6 ⊢ ((𝐴 = 𝑂 ∧ 𝑂 = 𝐶) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))
31		eqneqall 2951	. . . . . . . 8 ⊢ (𝐵 = 𝑇 → (𝐵 ≠ 𝑇 → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))
32	22, 31	mpi 20	. . . . . . 7 ⊢ (𝐵 = 𝑇 → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))
33	32	adantr 480	. . . . . 6 ⊢ ((𝐵 = 𝑇 ∧ 𝑇 = 𝐷) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))
34	27, 30, 33	ccase2 1040	. . . . 5 ⊢ ((((𝐴 = 𝐶 ∧ 𝑂 = 𝑂) ∨ (𝐴 = 𝑂 ∧ 𝑂 = 𝐶)) ∧ ((𝐵 = 𝐷 ∧ 𝑇 = 𝑇) ∨ (𝐵 = 𝑇 ∧ 𝑇 = 𝐷))) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))
35	21, 24, 34	syl2anb 598	. . . 4 ⊢ (({𝐴, 𝑂} = {𝐶, 𝑂} ∧ {𝐵, 𝑇} = {𝐷, 𝑇}) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))
36		preq12nebg 4863	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ 𝑂 ∈ V ∧ 𝐴 ≠ 𝑂) → ({𝐴, 𝑂} = {𝐷, 𝑇} ↔ ((𝐴 = 𝐷 ∧ 𝑂 = 𝑇) ∨ (𝐴 = 𝑇 ∧ 𝑂 = 𝐷))))
37	3, 4, 19, 36	mp3an 1463	. . . . 5 ⊢ ({𝐴, 𝑂} = {𝐷, 𝑇} ↔ ((𝐴 = 𝐷 ∧ 𝑂 = 𝑇) ∨ (𝐴 = 𝑇 ∧ 𝑂 = 𝐷)))
38		preq12nebg 4863	. . . . . 6 ⊢ ((𝐵 ∈ V ∧ 𝑇 ∈ V ∧ 𝐵 ≠ 𝑇) → ({𝐵, 𝑇} = {𝐶, 𝑂} ↔ ((𝐵 = 𝐶 ∧ 𝑇 = 𝑂) ∨ (𝐵 = 𝑂 ∧ 𝑇 = 𝐶))))
39	6, 7, 22, 38	mp3an 1463	. . . . 5 ⊢ ({𝐵, 𝑇} = {𝐶, 𝑂} ↔ ((𝐵 = 𝐶 ∧ 𝑇 = 𝑂) ∨ (𝐵 = 𝑂 ∧ 𝑇 = 𝐶)))
40		eqneqall 2951	. . . . . . . . . 10 ⊢ (𝑂 = 𝑇 → (𝑂 ≠ 𝑇 → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))
41	12, 40	mpi 20	. . . . . . . . 9 ⊢ (𝑂 = 𝑇 → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))
42	41	adantl 481	. . . . . . . 8 ⊢ ((𝐴 = 𝐷 ∧ 𝑂 = 𝑇) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))
43	42	a1d 25	. . . . . . 7 ⊢ ((𝐴 = 𝐷 ∧ 𝑂 = 𝑇) → (((𝐵 = 𝐶 ∧ 𝑇 = 𝑂) ∨ (𝐵 = 𝑂 ∧ 𝑇 = 𝐶)) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))
44	12	necomi 2995	. . . . . . . . . . . 12 ⊢ 𝑇 ≠ 𝑂
45		eqneqall 2951	. . . . . . . . . . . 12 ⊢ (𝑇 = 𝑂 → (𝑇 ≠ 𝑂 → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))
46	44, 45	mpi 20	. . . . . . . . . . 11 ⊢ (𝑇 = 𝑂 → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))
47	46	adantl 481	. . . . . . . . . 10 ⊢ ((𝐵 = 𝐶 ∧ 𝑇 = 𝑂) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))
48	47	a1d 25	. . . . . . . . 9 ⊢ ((𝐵 = 𝐶 ∧ 𝑇 = 𝑂) → ((𝐴 = 𝑇 ∧ 𝑂 = 𝐷) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))
49		eqneqall 2951	. . . . . . . . . . . 12 ⊢ (𝐵 = 𝑂 → (𝐵 ≠ 𝑂 → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))
50	10, 49	mpi 20	. . . . . . . . . . 11 ⊢ (𝐵 = 𝑂 → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))
51	50	adantr 480	. . . . . . . . . 10 ⊢ ((𝐵 = 𝑂 ∧ 𝑇 = 𝐶) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))
52	51	a1d 25	. . . . . . . . 9 ⊢ ((𝐵 = 𝑂 ∧ 𝑇 = 𝐶) → ((𝐴 = 𝑇 ∧ 𝑂 = 𝐷) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))
53	48, 52	jaoi 858	. . . . . . . 8 ⊢ (((𝐵 = 𝐶 ∧ 𝑇 = 𝑂) ∨ (𝐵 = 𝑂 ∧ 𝑇 = 𝐶)) → ((𝐴 = 𝑇 ∧ 𝑂 = 𝐷) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))
54	53	com12 32	. . . . . . 7 ⊢ ((𝐴 = 𝑇 ∧ 𝑂 = 𝐷) → (((𝐵 = 𝐶 ∧ 𝑇 = 𝑂) ∨ (𝐵 = 𝑂 ∧ 𝑇 = 𝐶)) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))
55	43, 54	jaoi 858	. . . . . 6 ⊢ (((𝐴 = 𝐷 ∧ 𝑂 = 𝑇) ∨ (𝐴 = 𝑇 ∧ 𝑂 = 𝐷)) → (((𝐵 = 𝐶 ∧ 𝑇 = 𝑂) ∨ (𝐵 = 𝑂 ∧ 𝑇 = 𝐶)) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))
56	55	imp 406	. . . . 5 ⊢ ((((𝐴 = 𝐷 ∧ 𝑂 = 𝑇) ∨ (𝐴 = 𝑇 ∧ 𝑂 = 𝐷)) ∧ ((𝐵 = 𝐶 ∧ 𝑇 = 𝑂) ∨ (𝐵 = 𝑂 ∧ 𝑇 = 𝐶))) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))
57	37, 39, 56	syl2anb 598	. . . 4 ⊢ (({𝐴, 𝑂} = {𝐷, 𝑇} ∧ {𝐵, 𝑇} = {𝐶, 𝑂}) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))
58	35, 57	jaoi 858	. . 3 ⊢ ((({𝐴, 𝑂} = {𝐶, 𝑂} ∧ {𝐵, 𝑇} = {𝐷, 𝑇}) ∨ ({𝐴, 𝑂} = {𝐷, 𝑇} ∧ {𝐵, 𝑇} = {𝐶, 𝑂})) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))
59	18, 58	sylbi 217	. 2 ⊢ ({{𝐴, 𝑂}, {𝐵, 𝑇}} = {{𝐶, 𝑂}, {𝐷, 𝑇}} → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))
60		preq1 4733	. . . 4 ⊢ (𝐴 = 𝐶 → {𝐴, 𝑂} = {𝐶, 𝑂})
61	60	adantr 480	. . 3 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → {𝐴, 𝑂} = {𝐶, 𝑂})
62		preq1 4733	. . . 4 ⊢ (𝐵 = 𝐷 → {𝐵, 𝑇} = {𝐷, 𝑇})
63	62	adantl 481	. . 3 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → {𝐵, 𝑇} = {𝐷, 𝑇})
64	61, 63	preq12d 4741	. 2 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → {{𝐴, 𝑂}, {𝐵, 𝑇}} = {{𝐶, 𝑂}, {𝐷, 𝑇}})
65	59, 64	impbii 209	1 ⊢ ({{𝐴, 𝑂}, {𝐵, 𝑇}} = {{𝐶, 𝑂}, {𝐷, 𝑇}} ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   = wceq 1540   ∈ wcel 2108   ≠ wne 2940  Vcvv 3480  {cpr 4628

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-v 3482  df-dif 3954  df-un 3956  df-nul 4334  df-sn 4627  df-pr 4629

This theorem is referenced by: (None)