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Theorem preleqg 9066
Description: Equality of two unordered pairs when one member of each pair contains the other member. Closed form of preleq 9067. (Contributed by AV, 15-Jun-2022.)
Assertion
Ref Expression
preleqg (((𝐴𝐵𝐵𝑉𝐶𝐷) ∧ {𝐴, 𝐵} = {𝐶, 𝐷}) → (𝐴 = 𝐶𝐵 = 𝐷))

Proof of Theorem preleqg
StepHypRef Expression
1 elneq 9050 . . . . 5 (𝐴𝐵𝐴𝐵)
213ad2ant1 1130 . . . 4 ((𝐴𝐵𝐵𝑉𝐶𝐷) → 𝐴𝐵)
3 preq12nebg 4766 . . . 4 ((𝐴𝐵𝐵𝑉𝐴𝐵) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶))))
42, 3syld3an3 1406 . . 3 ((𝐴𝐵𝐵𝑉𝐶𝐷) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶))))
5 eleq12 2903 . . . . . . . . . 10 ((𝐴 = 𝐷𝐵 = 𝐶) → (𝐴𝐵𝐷𝐶))
6 elnotel 9061 . . . . . . . . . . 11 (𝐷𝐶 → ¬ 𝐶𝐷)
76pm2.21d 121 . . . . . . . . . 10 (𝐷𝐶 → (𝐶𝐷 → (𝐴 = 𝐶𝐵 = 𝐷)))
85, 7syl6bi 256 . . . . . . . . 9 ((𝐴 = 𝐷𝐵 = 𝐶) → (𝐴𝐵 → (𝐶𝐷 → (𝐴 = 𝐶𝐵 = 𝐷))))
98com3l 89 . . . . . . . 8 (𝐴𝐵 → (𝐶𝐷 → ((𝐴 = 𝐷𝐵 = 𝐶) → (𝐴 = 𝐶𝐵 = 𝐷))))
109a1d 25 . . . . . . 7 (𝐴𝐵 → (𝐵𝑉 → (𝐶𝐷 → ((𝐴 = 𝐷𝐵 = 𝐶) → (𝐴 = 𝐶𝐵 = 𝐷)))))
11103imp 1108 . . . . . 6 ((𝐴𝐵𝐵𝑉𝐶𝐷) → ((𝐴 = 𝐷𝐵 = 𝐶) → (𝐴 = 𝐶𝐵 = 𝐷)))
1211com12 32 . . . . 5 ((𝐴 = 𝐷𝐵 = 𝐶) → ((𝐴𝐵𝐵𝑉𝐶𝐷) → (𝐴 = 𝐶𝐵 = 𝐷)))
1312jao1i 855 . . . 4 (((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶)) → ((𝐴𝐵𝐵𝑉𝐶𝐷) → (𝐴 = 𝐶𝐵 = 𝐷)))
1413com12 32 . . 3 ((𝐴𝐵𝐵𝑉𝐶𝐷) → (((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶)) → (𝐴 = 𝐶𝐵 = 𝐷)))
154, 14sylbid 243 . 2 ((𝐴𝐵𝐵𝑉𝐶𝐷) → ({𝐴, 𝐵} = {𝐶, 𝐷} → (𝐴 = 𝐶𝐵 = 𝐷)))
1615imp 410 1 (((𝐴𝐵𝐵𝑉𝐶𝐷) ∧ {𝐴, 𝐵} = {𝐶, 𝐷}) → (𝐴 = 𝐶𝐵 = 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wo 844  w3a 1084   = wceq 1538  wcel 2114  wne 3011  {cpr 4541
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 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307  ax-reg 9044
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-br 5043  df-opab 5105  df-eprel 5442  df-fr 5491
This theorem is referenced by:  preleq  9067
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