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Theorem opthhausdorff0 5376
Description: Justification theorem for the ordered pair definition of Felix Hausdorff in "Grundzüge der Mengenlehre" ("Basics of Set Theory"), 1914, p. 32: 𝐴, 𝐵H = {{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 5226). 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 5375. See df-op 4535 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 𝑂𝑇
Assertion
Ref Expression
opthhausdorff0 ({{𝐴, 𝑂}, {𝐵, 𝑇}} = {{𝐶, 𝑂}, {𝐷, 𝑇}} ↔ (𝐴 = 𝐶𝐵 = 𝐷))

Proof of Theorem opthhausdorff0
StepHypRef Expression
1 prex 5301 . . . 4 {𝐴, 𝑂} ∈ V
2 prex 5301 . . . 4 {𝐵, 𝑇} ∈ V
3 prex 5301 . . . 4 {𝐶, 𝑂} ∈ V
4 prex 5301 . . . 4 {𝐷, 𝑇} ∈ V
51, 2, 3, 4preq12b 4744 . . 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
86, 7preqr1 4742 . . . . 5 ({𝐴, 𝑂} = {𝐶, 𝑂} → 𝐴 = 𝐶)
9 opthhausdorff0.b . . . . . 6 𝐵 ∈ V
10 opthhausdorff0.d . . . . . 6 𝐷 ∈ V
119, 10preqr1 4742 . . . . 5 ({𝐵, 𝑇} = {𝐷, 𝑇} → 𝐵 = 𝐷)
128, 11anim12i 615 . . . 4 (({𝐴, 𝑂} = {𝐶, 𝑂} ∧ {𝐵, 𝑇} = {𝐷, 𝑇}) → (𝐴 = 𝐶𝐵 = 𝐷))
13 opthhausdorff0.1 . . . . . . 7 𝑂 ∈ V
14 opthhausdorff0.2 . . . . . . 7 𝑇 ∈ V
156, 13, 10, 14preq12b 4744 . . . . . 6 ({𝐴, 𝑂} = {𝐷, 𝑇} ↔ ((𝐴 = 𝐷𝑂 = 𝑇) ∨ (𝐴 = 𝑇𝑂 = 𝐷)))
16 opthhausdorff0.3 . . . . . . . . 9 𝑂𝑇
17 eqneqall 3001 . . . . . . . . 9 (𝑂 = 𝑇 → (𝑂𝑇 → ({𝐵, 𝑇} = {𝐶, 𝑂} → (𝐴 = 𝐶𝐵 = 𝐷))))
1816, 17mpi 20 . . . . . . . 8 (𝑂 = 𝑇 → ({𝐵, 𝑇} = {𝐶, 𝑂} → (𝐴 = 𝐶𝐵 = 𝐷)))
1918adantl 485 . . . . . . 7 ((𝐴 = 𝐷𝑂 = 𝑇) → ({𝐵, 𝑇} = {𝐶, 𝑂} → (𝐴 = 𝐶𝐵 = 𝐷)))
209, 14, 7, 13preq12b 4744 . . . . . . . . 9 ({𝐵, 𝑇} = {𝐶, 𝑂} ↔ ((𝐵 = 𝐶𝑇 = 𝑂) ∨ (𝐵 = 𝑂𝑇 = 𝐶)))
21 eqneqall 3001 . . . . . . . . . . . . 13 (𝑂 = 𝑇 → (𝑂𝑇 → ((𝐴 = 𝑇𝑂 = 𝐷) → (𝐴 = 𝐶𝐵 = 𝐷))))
2216, 21mpi 20 . . . . . . . . . . . 12 (𝑂 = 𝑇 → ((𝐴 = 𝑇𝑂 = 𝐷) → (𝐴 = 𝐶𝐵 = 𝐷)))
2322eqcoms 2809 . . . . . . . . . . 11 (𝑇 = 𝑂 → ((𝐴 = 𝑇𝑂 = 𝐷) → (𝐴 = 𝐶𝐵 = 𝐷)))
2423adantl 485 . . . . . . . . . 10 ((𝐵 = 𝐶𝑇 = 𝑂) → ((𝐴 = 𝑇𝑂 = 𝐷) → (𝐴 = 𝐶𝐵 = 𝐷)))
25 simpl 486 . . . . . . . . . . . . 13 ((𝐴 = 𝑇𝑂 = 𝐷) → 𝐴 = 𝑇)
26 simpr 488 . . . . . . . . . . . . 13 ((𝐵 = 𝑂𝑇 = 𝐶) → 𝑇 = 𝐶)
2725, 26sylan9eqr 2858 . . . . . . . . . . . 12 (((𝐵 = 𝑂𝑇 = 𝐶) ∧ (𝐴 = 𝑇𝑂 = 𝐷)) → 𝐴 = 𝐶)
28 simpl 486 . . . . . . . . . . . . 13 ((𝐵 = 𝑂𝑇 = 𝐶) → 𝐵 = 𝑂)
29 simpr 488 . . . . . . . . . . . . 13 ((𝐴 = 𝑇𝑂 = 𝐷) → 𝑂 = 𝐷)
3028, 29sylan9eq 2856 . . . . . . . . . . . 12 (((𝐵 = 𝑂𝑇 = 𝐶) ∧ (𝐴 = 𝑇𝑂 = 𝐷)) → 𝐵 = 𝐷)
3127, 30jca 515 . . . . . . . . . . 11 (((𝐵 = 𝑂𝑇 = 𝐶) ∧ (𝐴 = 𝑇𝑂 = 𝐷)) → (𝐴 = 𝐶𝐵 = 𝐷))
3231ex 416 . . . . . . . . . 10 ((𝐵 = 𝑂𝑇 = 𝐶) → ((𝐴 = 𝑇𝑂 = 𝐷) → (𝐴 = 𝐶𝐵 = 𝐷)))
3324, 32jaoi 854 . . . . . . . . 9 (((𝐵 = 𝐶𝑇 = 𝑂) ∨ (𝐵 = 𝑂𝑇 = 𝐶)) → ((𝐴 = 𝑇𝑂 = 𝐷) → (𝐴 = 𝐶𝐵 = 𝐷)))
3420, 33sylbi 220 . . . . . . . 8 ({𝐵, 𝑇} = {𝐶, 𝑂} → ((𝐴 = 𝑇𝑂 = 𝐷) → (𝐴 = 𝐶𝐵 = 𝐷)))
3534com12 32 . . . . . . 7 ((𝐴 = 𝑇𝑂 = 𝐷) → ({𝐵, 𝑇} = {𝐶, 𝑂} → (𝐴 = 𝐶𝐵 = 𝐷)))
3619, 35jaoi 854 . . . . . 6 (((𝐴 = 𝐷𝑂 = 𝑇) ∨ (𝐴 = 𝑇𝑂 = 𝐷)) → ({𝐵, 𝑇} = {𝐶, 𝑂} → (𝐴 = 𝐶𝐵 = 𝐷)))
3715, 36sylbi 220 . . . . 5 ({𝐴, 𝑂} = {𝐷, 𝑇} → ({𝐵, 𝑇} = {𝐶, 𝑂} → (𝐴 = 𝐶𝐵 = 𝐷)))
3837imp 410 . . . 4 (({𝐴, 𝑂} = {𝐷, 𝑇} ∧ {𝐵, 𝑇} = {𝐶, 𝑂}) → (𝐴 = 𝐶𝐵 = 𝐷))
3912, 38jaoi 854 . . 3 ((({𝐴, 𝑂} = {𝐶, 𝑂} ∧ {𝐵, 𝑇} = {𝐷, 𝑇}) ∨ ({𝐴, 𝑂} = {𝐷, 𝑇} ∧ {𝐵, 𝑇} = {𝐶, 𝑂})) → (𝐴 = 𝐶𝐵 = 𝐷))
405, 39sylbi 220 . 2 ({{𝐴, 𝑂}, {𝐵, 𝑇}} = {{𝐶, 𝑂}, {𝐷, 𝑇}} → (𝐴 = 𝐶𝐵 = 𝐷))
41 preq1 4632 . . . 4 (𝐴 = 𝐶 → {𝐴, 𝑂} = {𝐶, 𝑂})
4241adantr 484 . . 3 ((𝐴 = 𝐶𝐵 = 𝐷) → {𝐴, 𝑂} = {𝐶, 𝑂})
43 preq1 4632 . . . 4 (𝐵 = 𝐷 → {𝐵, 𝑇} = {𝐷, 𝑇})
4443adantl 485 . . 3 ((𝐴 = 𝐶𝐵 = 𝐷) → {𝐵, 𝑇} = {𝐷, 𝑇})
4542, 44preq12d 4640 . 2 ((𝐴 = 𝐶𝐵 = 𝐷) → {{𝐴, 𝑂}, {𝐵, 𝑇}} = {{𝐶, 𝑂}, {𝐷, 𝑇}})
4640, 45impbii 212 1 ({{𝐴, 𝑂}, {𝐵, 𝑇}} = {{𝐶, 𝑂}, {𝐷, 𝑇}} ↔ (𝐴 = 𝐶𝐵 = 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wo 844   = wceq 1538  wcel 2112  wne 2990  Vcvv 3444  {cpr 4530
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-ne 2991  df-v 3446  df-dif 3887  df-un 3889  df-nul 4247  df-sn 4529  df-pr 4531
This theorem is referenced by: (None)
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