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Theorem preleqg 9653
Description: Equality of two unordered pairs when one member of each pair contains the other member. Closed form of preleq 9654. (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 9636 . . . . 5 (𝐴𝐵𝐴𝐵)
213ad2ant1 1132 . . . 4 ((𝐴𝐵𝐵𝑉𝐶𝐷) → 𝐴𝐵)
3 preq12nebg 4868 . . . 4 ((𝐴𝐵𝐵𝑉𝐴𝐵) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶))))
42, 3syld3an3 1408 . . 3 ((𝐴𝐵𝐵𝑉𝐶𝐷) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶))))
5 eleq12 2829 . . . . . . . . . 10 ((𝐴 = 𝐷𝐵 = 𝐶) → (𝐴𝐵𝐷𝐶))
6 elnotel 9648 . . . . . . . . . . 11 (𝐷𝐶 → ¬ 𝐶𝐷)
76pm2.21d 121 . . . . . . . . . 10 (𝐷𝐶 → (𝐶𝐷 → (𝐴 = 𝐶𝐵 = 𝐷)))
85, 7biimtrdi 253 . . . . . . . . 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 1110 . . . . . 6 ((𝐴𝐵𝐵𝑉𝐶𝐷) → ((𝐴 = 𝐷𝐵 = 𝐶) → (𝐴 = 𝐶𝐵 = 𝐷)))
1211com12 32 . . . . 5 ((𝐴 = 𝐷𝐵 = 𝐶) → ((𝐴𝐵𝐵𝑉𝐶𝐷) → (𝐴 = 𝐶𝐵 = 𝐷)))
1312jao1i 858 . . . 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 240 . 2 ((𝐴𝐵𝐵𝑉𝐶𝐷) → ({𝐴, 𝐵} = {𝐶, 𝐷} → (𝐴 = 𝐶𝐵 = 𝐷)))
1615imp 406 1 (((𝐴𝐵𝐵𝑉𝐶𝐷) ∧ {𝐴, 𝐵} = {𝐶, 𝐷}) → (𝐴 = 𝐶𝐵 = 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wo 847  w3a 1086   = wceq 1537  wcel 2106  wne 2938  {cpr 4633
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-reg 9630
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-eprel 5589  df-fr 5641
This theorem is referenced by:  preleq  9654
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