MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  preleqg Structured version   Visualization version   GIF version

Theorem preleqg 9610
Description: Equality of two unordered pairs when one member of each pair contains the other member. Closed form of preleq 9611. (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 9593 . . . . 5 (𝐴𝐵𝐴𝐵)
213ad2ant1 1134 . . . 4 ((𝐴𝐵𝐵𝑉𝐶𝐷) → 𝐴𝐵)
3 preq12nebg 4864 . . . 4 ((𝐴𝐵𝐵𝑉𝐴𝐵) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶))))
42, 3syld3an3 1410 . . 3 ((𝐴𝐵𝐵𝑉𝐶𝐷) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶))))
5 eleq12 2824 . . . . . . . . . 10 ((𝐴 = 𝐷𝐵 = 𝐶) → (𝐴𝐵𝐷𝐶))
6 elnotel 9605 . . . . . . . . . . 11 (𝐷𝐶 → ¬ 𝐶𝐷)
76pm2.21d 121 . . . . . . . . . 10 (𝐷𝐶 → (𝐶𝐷 → (𝐴 = 𝐶𝐵 = 𝐷)))
85, 7syl6bi 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 1112 . . . . . 6 ((𝐴𝐵𝐵𝑉𝐶𝐷) → ((𝐴 = 𝐷𝐵 = 𝐶) → (𝐴 = 𝐶𝐵 = 𝐷)))
1211com12 32 . . . . 5 ((𝐴 = 𝐷𝐵 = 𝐶) → ((𝐴𝐵𝐵𝑉𝐶𝐷) → (𝐴 = 𝐶𝐵 = 𝐷)))
1312jao1i 857 . . . 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 408 1 (((𝐴𝐵𝐵𝑉𝐶𝐷) ∧ {𝐴, 𝐵} = {𝐶, 𝐷}) → (𝐴 = 𝐶𝐵 = 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wo 846  w3a 1088   = wceq 1542  wcel 2107  wne 2941  {cpr 4631
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-10 2138  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-reg 9587
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-eprel 5581  df-fr 5632
This theorem is referenced by:  preleq  9611
  Copyright terms: Public domain W3C validator