Theorem opthhausdorff0 5432

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 5280). Furthermore, Hausdorff demanded that 𝑂 and 𝑇 are both different from 𝐴 as well as 𝐵, which is actually not necessary if all involved classes exist as sets (i.e. are not proper classes), in contrast to opthhausdorff 5431. See df-op 4568 for other ordered pair definitions. (Contributed by AV, 12-Jun-2022.)

Hypotheses

Ref	Expression
opthhausdorff0.a	⊢ 𝐴 ∈ V
opthhausdorff0.b	⊢ 𝐵 ∈ V
opthhausdorff0.c	⊢ 𝐶 ∈ V
opthhausdorff0.d	⊢ 𝐷 ∈ V
opthhausdorff0.1	⊢ 𝑂 ∈ V
opthhausdorff0.2	⊢ 𝑇 ∈ V
opthhausdorff0.3	⊢ 𝑂 ≠ 𝑇

Assertion

Ref	Expression
opthhausdorff0	⊢ ({{𝐴, 𝑂}, {𝐵, 𝑇}} = {{𝐶, 𝑂}, {𝐷, 𝑇}} ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))

Proof of Theorem opthhausdorff0

Step	Hyp	Ref	Expression
1		prex 5355	. . . 4 ⊢ {𝐴, 𝑂} ∈ V
2		prex 5355	. . . 4 ⊢ {𝐵, 𝑇} ∈ V
3		prex 5355	. . . 4 ⊢ {𝐶, 𝑂} ∈ V
4		prex 5355	. . . 4 ⊢ {𝐷, 𝑇} ∈ V
5	1, 2, 3, 4	preq12b 4781	. . 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})))
6		opthhausdorff0.a	. . . . . 6 ⊢ 𝐴 ∈ V
7		opthhausdorff0.c	. . . . . 6 ⊢ 𝐶 ∈ V
8	6, 7	preqr1 4779	. . . . 5 ⊢ ({𝐴, 𝑂} = {𝐶, 𝑂} → 𝐴 = 𝐶)
9		opthhausdorff0.b	. . . . . 6 ⊢ 𝐵 ∈ V
10		opthhausdorff0.d	. . . . . 6 ⊢ 𝐷 ∈ V
11	9, 10	preqr1 4779	. . . . 5 ⊢ ({𝐵, 𝑇} = {𝐷, 𝑇} → 𝐵 = 𝐷)
12	8, 11	anim12i 613	. . . 4 ⊢ (({𝐴, 𝑂} = {𝐶, 𝑂} ∧ {𝐵, 𝑇} = {𝐷, 𝑇}) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))
13		opthhausdorff0.1	. . . . . . 7 ⊢ 𝑂 ∈ V
14		opthhausdorff0.2	. . . . . . 7 ⊢ 𝑇 ∈ V
15	6, 13, 10, 14	preq12b 4781	. . . . . 6 ⊢ ({𝐴, 𝑂} = {𝐷, 𝑇} ↔ ((𝐴 = 𝐷 ∧ 𝑂 = 𝑇) ∨ (𝐴 = 𝑇 ∧ 𝑂 = 𝐷)))
16		opthhausdorff0.3	. . . . . . . . 9 ⊢ 𝑂 ≠ 𝑇
17		eqneqall 2954	. . . . . . . . 9 ⊢ (𝑂 = 𝑇 → (𝑂 ≠ 𝑇 → ({𝐵, 𝑇} = {𝐶, 𝑂} → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))))
18	16, 17	mpi 20	. . . . . . . 8 ⊢ (𝑂 = 𝑇 → ({𝐵, 𝑇} = {𝐶, 𝑂} → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))
19	18	adantl 482	. . . . . . 7 ⊢ ((𝐴 = 𝐷 ∧ 𝑂 = 𝑇) → ({𝐵, 𝑇} = {𝐶, 𝑂} → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))
20	9, 14, 7, 13	preq12b 4781	. . . . . . . . 9 ⊢ ({𝐵, 𝑇} = {𝐶, 𝑂} ↔ ((𝐵 = 𝐶 ∧ 𝑇 = 𝑂) ∨ (𝐵 = 𝑂 ∧ 𝑇 = 𝐶)))
21		eqneqall 2954	. . . . . . . . . . . . 13 ⊢ (𝑂 = 𝑇 → (𝑂 ≠ 𝑇 → ((𝐴 = 𝑇 ∧ 𝑂 = 𝐷) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))))
22	16, 21	mpi 20	. . . . . . . . . . . 12 ⊢ (𝑂 = 𝑇 → ((𝐴 = 𝑇 ∧ 𝑂 = 𝐷) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))
23	22	eqcoms 2746	. . . . . . . . . . 11 ⊢ (𝑇 = 𝑂 → ((𝐴 = 𝑇 ∧ 𝑂 = 𝐷) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))
24	23	adantl 482	. . . . . . . . . 10 ⊢ ((𝐵 = 𝐶 ∧ 𝑇 = 𝑂) → ((𝐴 = 𝑇 ∧ 𝑂 = 𝐷) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))
25		simpl 483	. . . . . . . . . . . . 13 ⊢ ((𝐴 = 𝑇 ∧ 𝑂 = 𝐷) → 𝐴 = 𝑇)
26		simpr 485	. . . . . . . . . . . . 13 ⊢ ((𝐵 = 𝑂 ∧ 𝑇 = 𝐶) → 𝑇 = 𝐶)
27	25, 26	sylan9eqr 2800	. . . . . . . . . . . 12 ⊢ (((𝐵 = 𝑂 ∧ 𝑇 = 𝐶) ∧ (𝐴 = 𝑇 ∧ 𝑂 = 𝐷)) → 𝐴 = 𝐶)
28		simpl 483	. . . . . . . . . . . . 13 ⊢ ((𝐵 = 𝑂 ∧ 𝑇 = 𝐶) → 𝐵 = 𝑂)
29		simpr 485	. . . . . . . . . . . . 13 ⊢ ((𝐴 = 𝑇 ∧ 𝑂 = 𝐷) → 𝑂 = 𝐷)
30	28, 29	sylan9eq 2798	. . . . . . . . . . . 12 ⊢ (((𝐵 = 𝑂 ∧ 𝑇 = 𝐶) ∧ (𝐴 = 𝑇 ∧ 𝑂 = 𝐷)) → 𝐵 = 𝐷)
31	27, 30	jca 512	. . . . . . . . . . 11 ⊢ (((𝐵 = 𝑂 ∧ 𝑇 = 𝐶) ∧ (𝐴 = 𝑇 ∧ 𝑂 = 𝐷)) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))
32	31	ex 413	. . . . . . . . . 10 ⊢ ((𝐵 = 𝑂 ∧ 𝑇 = 𝐶) → ((𝐴 = 𝑇 ∧ 𝑂 = 𝐷) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))
33	24, 32	jaoi 854	. . . . . . . . 9 ⊢ (((𝐵 = 𝐶 ∧ 𝑇 = 𝑂) ∨ (𝐵 = 𝑂 ∧ 𝑇 = 𝐶)) → ((𝐴 = 𝑇 ∧ 𝑂 = 𝐷) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))
34	20, 33	sylbi 216	. . . . . . . 8 ⊢ ({𝐵, 𝑇} = {𝐶, 𝑂} → ((𝐴 = 𝑇 ∧ 𝑂 = 𝐷) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))
35	34	com12 32	. . . . . . 7 ⊢ ((𝐴 = 𝑇 ∧ 𝑂 = 𝐷) → ({𝐵, 𝑇} = {𝐶, 𝑂} → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))
36	19, 35	jaoi 854	. . . . . 6 ⊢ (((𝐴 = 𝐷 ∧ 𝑂 = 𝑇) ∨ (𝐴 = 𝑇 ∧ 𝑂 = 𝐷)) → ({𝐵, 𝑇} = {𝐶, 𝑂} → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))
37	15, 36	sylbi 216	. . . . 5 ⊢ ({𝐴, 𝑂} = {𝐷, 𝑇} → ({𝐵, 𝑇} = {𝐶, 𝑂} → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))
38	37	imp 407	. . . 4 ⊢ (({𝐴, 𝑂} = {𝐷, 𝑇} ∧ {𝐵, 𝑇} = {𝐶, 𝑂}) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))
39	12, 38	jaoi 854	. . 3 ⊢ ((({𝐴, 𝑂} = {𝐶, 𝑂} ∧ {𝐵, 𝑇} = {𝐷, 𝑇}) ∨ ({𝐴, 𝑂} = {𝐷, 𝑇} ∧ {𝐵, 𝑇} = {𝐶, 𝑂})) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))
40	5, 39	sylbi 216	. 2 ⊢ ({{𝐴, 𝑂}, {𝐵, 𝑇}} = {{𝐶, 𝑂}, {𝐷, 𝑇}} → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))
41		preq1 4669	. . . 4 ⊢ (𝐴 = 𝐶 → {𝐴, 𝑂} = {𝐶, 𝑂})
42	41	adantr 481	. . 3 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → {𝐴, 𝑂} = {𝐶, 𝑂})
43		preq1 4669	. . . 4 ⊢ (𝐵 = 𝐷 → {𝐵, 𝑇} = {𝐷, 𝑇})
44	43	adantl 482	. . 3 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → {𝐵, 𝑇} = {𝐷, 𝑇})
45	42, 44	preq12d 4677	. 2 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → {{𝐴, 𝑂}, {𝐵, 𝑇}} = {{𝐶, 𝑂}, {𝐷, 𝑇}})
46	40, 45	impbii 208	1 ⊢ ({{𝐴, 𝑂}, {𝐵, 𝑇}} = {{𝐶, 𝑂}, {𝐷, 𝑇}} ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 844   = wceq 1539   ∈ wcel 2106   ≠ wne 2943  Vcvv 3432  {cpr 4563

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-v 3434  df-dif 3890  df-un 3892  df-nul 4257  df-sn 4562  df-pr 4564

This theorem is referenced by: (None)