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Theorem opthhausdorff0 5476
Description: Justification theorem for the ordered pair definition of Felix Hausdorff in "GrundzΓΌge der Mengenlehre" ("Basics of Set Theory"), 1914, p. 32: ⟨𝐴, 𝐡⟩H = {{𝐴, 𝑂}, {𝐡, 𝑇}}. Hausdorff used 1 and 2 instead of 𝑂 and 𝑇, but actually, any two different fixed sets will do (e.g., 𝑂 = βˆ… and 𝑇 = {βˆ…}, see 0nep0 5314). Furthermore, Hausdorff demanded that 𝑂 and 𝑇 are both different from 𝐴 as well as 𝐡, which is actually not necessary if all involved classes exist as sets (i.e. are not proper classes), in contrast to opthhausdorff 5475. See df-op 4594 for other ordered pair definitions. (Contributed by AV, 12-Jun-2022.)
Hypotheses
Ref Expression
opthhausdorff0.a 𝐴 ∈ V
opthhausdorff0.b 𝐡 ∈ V
opthhausdorff0.c 𝐢 ∈ V
opthhausdorff0.d 𝐷 ∈ V
opthhausdorff0.1 𝑂 ∈ V
opthhausdorff0.2 𝑇 ∈ V
opthhausdorff0.3 𝑂 β‰  𝑇
Assertion
Ref Expression
opthhausdorff0 ({{𝐴, 𝑂}, {𝐡, 𝑇}} = {{𝐢, 𝑂}, {𝐷, 𝑇}} ↔ (𝐴 = 𝐢 ∧ 𝐡 = 𝐷))

Proof of Theorem opthhausdorff0
StepHypRef Expression
1 prex 5390 . . . 4 {𝐴, 𝑂} ∈ V
2 prex 5390 . . . 4 {𝐡, 𝑇} ∈ V
3 prex 5390 . . . 4 {𝐢, 𝑂} ∈ V
4 prex 5390 . . . 4 {𝐷, 𝑇} ∈ V
51, 2, 3, 4preq12b 4809 . . 3 ({{𝐴, 𝑂}, {𝐡, 𝑇}} = {{𝐢, 𝑂}, {𝐷, 𝑇}} ↔ (({𝐴, 𝑂} = {𝐢, 𝑂} ∧ {𝐡, 𝑇} = {𝐷, 𝑇}) ∨ ({𝐴, 𝑂} = {𝐷, 𝑇} ∧ {𝐡, 𝑇} = {𝐢, 𝑂})))
6 opthhausdorff0.a . . . . . 6 𝐴 ∈ V
7 opthhausdorff0.c . . . . . 6 𝐢 ∈ V
86, 7preqr1 4807 . . . . 5 ({𝐴, 𝑂} = {𝐢, 𝑂} β†’ 𝐴 = 𝐢)
9 opthhausdorff0.b . . . . . 6 𝐡 ∈ V
10 opthhausdorff0.d . . . . . 6 𝐷 ∈ V
119, 10preqr1 4807 . . . . 5 ({𝐡, 𝑇} = {𝐷, 𝑇} β†’ 𝐡 = 𝐷)
128, 11anim12i 614 . . . 4 (({𝐴, 𝑂} = {𝐢, 𝑂} ∧ {𝐡, 𝑇} = {𝐷, 𝑇}) β†’ (𝐴 = 𝐢 ∧ 𝐡 = 𝐷))
13 opthhausdorff0.1 . . . . . . 7 𝑂 ∈ V
14 opthhausdorff0.2 . . . . . . 7 𝑇 ∈ V
156, 13, 10, 14preq12b 4809 . . . . . 6 ({𝐴, 𝑂} = {𝐷, 𝑇} ↔ ((𝐴 = 𝐷 ∧ 𝑂 = 𝑇) ∨ (𝐴 = 𝑇 ∧ 𝑂 = 𝐷)))
16 opthhausdorff0.3 . . . . . . . . 9 𝑂 β‰  𝑇
17 eqneqall 2951 . . . . . . . . 9 (𝑂 = 𝑇 β†’ (𝑂 β‰  𝑇 β†’ ({𝐡, 𝑇} = {𝐢, 𝑂} β†’ (𝐴 = 𝐢 ∧ 𝐡 = 𝐷))))
1816, 17mpi 20 . . . . . . . 8 (𝑂 = 𝑇 β†’ ({𝐡, 𝑇} = {𝐢, 𝑂} β†’ (𝐴 = 𝐢 ∧ 𝐡 = 𝐷)))
1918adantl 483 . . . . . . 7 ((𝐴 = 𝐷 ∧ 𝑂 = 𝑇) β†’ ({𝐡, 𝑇} = {𝐢, 𝑂} β†’ (𝐴 = 𝐢 ∧ 𝐡 = 𝐷)))
209, 14, 7, 13preq12b 4809 . . . . . . . . 9 ({𝐡, 𝑇} = {𝐢, 𝑂} ↔ ((𝐡 = 𝐢 ∧ 𝑇 = 𝑂) ∨ (𝐡 = 𝑂 ∧ 𝑇 = 𝐢)))
21 eqneqall 2951 . . . . . . . . . . . . 13 (𝑂 = 𝑇 β†’ (𝑂 β‰  𝑇 β†’ ((𝐴 = 𝑇 ∧ 𝑂 = 𝐷) β†’ (𝐴 = 𝐢 ∧ 𝐡 = 𝐷))))
2216, 21mpi 20 . . . . . . . . . . . 12 (𝑂 = 𝑇 β†’ ((𝐴 = 𝑇 ∧ 𝑂 = 𝐷) β†’ (𝐴 = 𝐢 ∧ 𝐡 = 𝐷)))
2322eqcoms 2741 . . . . . . . . . . 11 (𝑇 = 𝑂 β†’ ((𝐴 = 𝑇 ∧ 𝑂 = 𝐷) β†’ (𝐴 = 𝐢 ∧ 𝐡 = 𝐷)))
2423adantl 483 . . . . . . . . . 10 ((𝐡 = 𝐢 ∧ 𝑇 = 𝑂) β†’ ((𝐴 = 𝑇 ∧ 𝑂 = 𝐷) β†’ (𝐴 = 𝐢 ∧ 𝐡 = 𝐷)))
25 simpl 484 . . . . . . . . . . . . 13 ((𝐴 = 𝑇 ∧ 𝑂 = 𝐷) β†’ 𝐴 = 𝑇)
26 simpr 486 . . . . . . . . . . . . 13 ((𝐡 = 𝑂 ∧ 𝑇 = 𝐢) β†’ 𝑇 = 𝐢)
2725, 26sylan9eqr 2795 . . . . . . . . . . . 12 (((𝐡 = 𝑂 ∧ 𝑇 = 𝐢) ∧ (𝐴 = 𝑇 ∧ 𝑂 = 𝐷)) β†’ 𝐴 = 𝐢)
28 simpl 484 . . . . . . . . . . . . 13 ((𝐡 = 𝑂 ∧ 𝑇 = 𝐢) β†’ 𝐡 = 𝑂)
29 simpr 486 . . . . . . . . . . . . 13 ((𝐴 = 𝑇 ∧ 𝑂 = 𝐷) β†’ 𝑂 = 𝐷)
3028, 29sylan9eq 2793 . . . . . . . . . . . 12 (((𝐡 = 𝑂 ∧ 𝑇 = 𝐢) ∧ (𝐴 = 𝑇 ∧ 𝑂 = 𝐷)) β†’ 𝐡 = 𝐷)
3127, 30jca 513 . . . . . . . . . . 11 (((𝐡 = 𝑂 ∧ 𝑇 = 𝐢) ∧ (𝐴 = 𝑇 ∧ 𝑂 = 𝐷)) β†’ (𝐴 = 𝐢 ∧ 𝐡 = 𝐷))
3231ex 414 . . . . . . . . . 10 ((𝐡 = 𝑂 ∧ 𝑇 = 𝐢) β†’ ((𝐴 = 𝑇 ∧ 𝑂 = 𝐷) β†’ (𝐴 = 𝐢 ∧ 𝐡 = 𝐷)))
3324, 32jaoi 856 . . . . . . . . 9 (((𝐡 = 𝐢 ∧ 𝑇 = 𝑂) ∨ (𝐡 = 𝑂 ∧ 𝑇 = 𝐢)) β†’ ((𝐴 = 𝑇 ∧ 𝑂 = 𝐷) β†’ (𝐴 = 𝐢 ∧ 𝐡 = 𝐷)))
3420, 33sylbi 216 . . . . . . . 8 ({𝐡, 𝑇} = {𝐢, 𝑂} β†’ ((𝐴 = 𝑇 ∧ 𝑂 = 𝐷) β†’ (𝐴 = 𝐢 ∧ 𝐡 = 𝐷)))
3534com12 32 . . . . . . 7 ((𝐴 = 𝑇 ∧ 𝑂 = 𝐷) β†’ ({𝐡, 𝑇} = {𝐢, 𝑂} β†’ (𝐴 = 𝐢 ∧ 𝐡 = 𝐷)))
3619, 35jaoi 856 . . . . . 6 (((𝐴 = 𝐷 ∧ 𝑂 = 𝑇) ∨ (𝐴 = 𝑇 ∧ 𝑂 = 𝐷)) β†’ ({𝐡, 𝑇} = {𝐢, 𝑂} β†’ (𝐴 = 𝐢 ∧ 𝐡 = 𝐷)))
3715, 36sylbi 216 . . . . 5 ({𝐴, 𝑂} = {𝐷, 𝑇} β†’ ({𝐡, 𝑇} = {𝐢, 𝑂} β†’ (𝐴 = 𝐢 ∧ 𝐡 = 𝐷)))
3837imp 408 . . . 4 (({𝐴, 𝑂} = {𝐷, 𝑇} ∧ {𝐡, 𝑇} = {𝐢, 𝑂}) β†’ (𝐴 = 𝐢 ∧ 𝐡 = 𝐷))
3912, 38jaoi 856 . . 3 ((({𝐴, 𝑂} = {𝐢, 𝑂} ∧ {𝐡, 𝑇} = {𝐷, 𝑇}) ∨ ({𝐴, 𝑂} = {𝐷, 𝑇} ∧ {𝐡, 𝑇} = {𝐢, 𝑂})) β†’ (𝐴 = 𝐢 ∧ 𝐡 = 𝐷))
405, 39sylbi 216 . 2 ({{𝐴, 𝑂}, {𝐡, 𝑇}} = {{𝐢, 𝑂}, {𝐷, 𝑇}} β†’ (𝐴 = 𝐢 ∧ 𝐡 = 𝐷))
41 preq1 4695 . . . 4 (𝐴 = 𝐢 β†’ {𝐴, 𝑂} = {𝐢, 𝑂})
4241adantr 482 . . 3 ((𝐴 = 𝐢 ∧ 𝐡 = 𝐷) β†’ {𝐴, 𝑂} = {𝐢, 𝑂})
43 preq1 4695 . . . 4 (𝐡 = 𝐷 β†’ {𝐡, 𝑇} = {𝐷, 𝑇})
4443adantl 483 . . 3 ((𝐴 = 𝐢 ∧ 𝐡 = 𝐷) β†’ {𝐡, 𝑇} = {𝐷, 𝑇})
4542, 44preq12d 4703 . 2 ((𝐴 = 𝐢 ∧ 𝐡 = 𝐷) β†’ {{𝐴, 𝑂}, {𝐡, 𝑇}} = {{𝐢, 𝑂}, {𝐷, 𝑇}})
4640, 45impbii 208 1 ({{𝐴, 𝑂}, {𝐡, 𝑇}} = {{𝐢, 𝑂}, {𝐷, 𝑇}} ↔ (𝐴 = 𝐢 ∧ 𝐡 = 𝐷))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∨ wo 846   = wceq 1542   ∈ wcel 2107   β‰  wne 2940  Vcvv 3444  {cpr 4589
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-v 3446  df-dif 3914  df-un 3916  df-nul 4284  df-sn 4588  df-pr 4590
This theorem is referenced by: (None)
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