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Theorem preleqg 9551
Description: Equality of two unordered pairs when one member of each pair contains the other member. Closed form of preleq 9552. (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 9534 . . . . 5 (𝐴𝐵𝐴𝐵)
213ad2ant1 1133 . . . 4 ((𝐴𝐵𝐵𝑉𝐶𝐷) → 𝐴𝐵)
3 preq12nebg 4820 . . . 4 ((𝐴𝐵𝐵𝑉𝐴𝐵) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶))))
42, 3syld3an3 1409 . . 3 ((𝐴𝐵𝐵𝑉𝐶𝐷) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶))))
5 eleq12 2827 . . . . . . . . . 10 ((𝐴 = 𝐷𝐵 = 𝐶) → (𝐴𝐵𝐷𝐶))
6 elnotel 9546 . . . . . . . . . . 11 (𝐷𝐶 → ¬ 𝐶𝐷)
76pm2.21d 121 . . . . . . . . . 10 (𝐷𝐶 → (𝐶𝐷 → (𝐴 = 𝐶𝐵 = 𝐷)))
85, 7syl6bi 252 . . . . . . . . 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 1111 . . . . . 6 ((𝐴𝐵𝐵𝑉𝐶𝐷) → ((𝐴 = 𝐷𝐵 = 𝐶) → (𝐴 = 𝐶𝐵 = 𝐷)))
1211com12 32 . . . . 5 ((𝐴 = 𝐷𝐵 = 𝐶) → ((𝐴𝐵𝐵𝑉𝐶𝐷) → (𝐴 = 𝐶𝐵 = 𝐷)))
1312jao1i 856 . . . 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 239 . 2 ((𝐴𝐵𝐵𝑉𝐶𝐷) → ({𝐴, 𝐵} = {𝐶, 𝐷} → (𝐴 = 𝐶𝐵 = 𝐷)))
1615imp 407 1 (((𝐴𝐵𝐵𝑉𝐶𝐷) ∧ {𝐴, 𝐵} = {𝐶, 𝐷}) → (𝐴 = 𝐶𝐵 = 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wo 845  w3a 1087   = wceq 1541  wcel 2106  wne 2943  {cpr 4588
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-reg 9528
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-br 5106  df-opab 5168  df-eprel 5537  df-fr 5588
This theorem is referenced by:  preleq  9552
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