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Theorem poprelb 44028
Description: Equality for unordered pairs with partially ordered elements. (Contributed by AV, 9-Jul-2023.)
Assertion
Ref Expression
poprelb (((Rel 𝑅𝑅 Po 𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐶𝑅𝐷)) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ (𝐴 = 𝐶𝐵 = 𝐷)))

Proof of Theorem poprelb
StepHypRef Expression
1 simp2 1134 . . 3 (((Rel 𝑅𝑅 Po 𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐶𝑅𝐷)) → (𝐴𝑋𝐵𝑋))
2 an3 658 . . . . 5 (((Rel 𝑅𝑅 Po 𝑋) ∧ (𝐴𝑅𝐵𝐶𝑅𝐷)) → (Rel 𝑅𝐶𝑅𝐷))
323adant2 1128 . . . 4 (((Rel 𝑅𝑅 Po 𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐶𝑅𝐷)) → (Rel 𝑅𝐶𝑅𝐷))
4 brrelex12 5572 . . . 4 ((Rel 𝑅𝐶𝑅𝐷) → (𝐶 ∈ V ∧ 𝐷 ∈ V))
53, 4syl 17 . . 3 (((Rel 𝑅𝑅 Po 𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐶𝑅𝐷)) → (𝐶 ∈ V ∧ 𝐷 ∈ V))
6 preq12bg 4747 . . 3 (((𝐴𝑋𝐵𝑋) ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶))))
71, 5, 6syl2anc 587 . 2 (((Rel 𝑅𝑅 Po 𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐶𝑅𝐷)) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶))))
8 idd 24 . . . 4 (((Rel 𝑅𝑅 Po 𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐶𝑅𝐷)) → ((𝐴 = 𝐶𝐵 = 𝐷) → (𝐴 = 𝐶𝐵 = 𝐷)))
9 breq12 5038 . . . . . . . . . . 11 ((𝐵 = 𝐶𝐴 = 𝐷) → (𝐵𝑅𝐴𝐶𝑅𝐷))
109ancoms 462 . . . . . . . . . 10 ((𝐴 = 𝐷𝐵 = 𝐶) → (𝐵𝑅𝐴𝐶𝑅𝐷))
1110bicomd 226 . . . . . . . . 9 ((𝐴 = 𝐷𝐵 = 𝐶) → (𝐶𝑅𝐷𝐵𝑅𝐴))
1211anbi2d 631 . . . . . . . 8 ((𝐴 = 𝐷𝐵 = 𝐶) → ((𝐴𝑅𝐵𝐶𝑅𝐷) ↔ (𝐴𝑅𝐵𝐵𝑅𝐴)))
13 po2nr 5455 . . . . . . . . . . . 12 ((𝑅 Po 𝑋 ∧ (𝐴𝑋𝐵𝑋)) → ¬ (𝐴𝑅𝐵𝐵𝑅𝐴))
1413adantll 713 . . . . . . . . . . 11 (((Rel 𝑅𝑅 Po 𝑋) ∧ (𝐴𝑋𝐵𝑋)) → ¬ (𝐴𝑅𝐵𝐵𝑅𝐴))
1514pm2.21d 121 . . . . . . . . . 10 (((Rel 𝑅𝑅 Po 𝑋) ∧ (𝐴𝑋𝐵𝑋)) → ((𝐴𝑅𝐵𝐵𝑅𝐴) → (𝐴 = 𝐶𝐵 = 𝐷)))
1615ex 416 . . . . . . . . 9 ((Rel 𝑅𝑅 Po 𝑋) → ((𝐴𝑋𝐵𝑋) → ((𝐴𝑅𝐵𝐵𝑅𝐴) → (𝐴 = 𝐶𝐵 = 𝐷))))
1716com13 88 . . . . . . . 8 ((𝐴𝑅𝐵𝐵𝑅𝐴) → ((𝐴𝑋𝐵𝑋) → ((Rel 𝑅𝑅 Po 𝑋) → (𝐴 = 𝐶𝐵 = 𝐷))))
1812, 17syl6bi 256 . . . . . . 7 ((𝐴 = 𝐷𝐵 = 𝐶) → ((𝐴𝑅𝐵𝐶𝑅𝐷) → ((𝐴𝑋𝐵𝑋) → ((Rel 𝑅𝑅 Po 𝑋) → (𝐴 = 𝐶𝐵 = 𝐷)))))
1918com23 86 . . . . . 6 ((𝐴 = 𝐷𝐵 = 𝐶) → ((𝐴𝑋𝐵𝑋) → ((𝐴𝑅𝐵𝐶𝑅𝐷) → ((Rel 𝑅𝑅 Po 𝑋) → (𝐴 = 𝐶𝐵 = 𝐷)))))
2019com14 96 . . . . 5 ((Rel 𝑅𝑅 Po 𝑋) → ((𝐴𝑋𝐵𝑋) → ((𝐴𝑅𝐵𝐶𝑅𝐷) → ((𝐴 = 𝐷𝐵 = 𝐶) → (𝐴 = 𝐶𝐵 = 𝐷)))))
21203imp 1108 . . . 4 (((Rel 𝑅𝑅 Po 𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐶𝑅𝐷)) → ((𝐴 = 𝐷𝐵 = 𝐶) → (𝐴 = 𝐶𝐵 = 𝐷)))
228, 21jaod 856 . . 3 (((Rel 𝑅𝑅 Po 𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐶𝑅𝐷)) → (((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶)) → (𝐴 = 𝐶𝐵 = 𝐷)))
23 orc 864 . . 3 ((𝐴 = 𝐶𝐵 = 𝐷) → ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶)))
2422, 23impbid1 228 . 2 (((Rel 𝑅𝑅 Po 𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐶𝑅𝐷)) → (((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶)) ↔ (𝐴 = 𝐶𝐵 = 𝐷)))
257, 24bitrd 282 1 (((Rel 𝑅𝑅 Po 𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐶𝑅𝐷)) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ (𝐴 = 𝐶𝐵 = 𝐷)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wo 844  w3a 1084   = wceq 1538  wcel 2112  Vcvv 3444  {cpr 4530   class class class wbr 5033   Po wpo 5440  Rel wrel 5528
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 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298
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-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-br 5034  df-opab 5096  df-po 5442  df-xp 5529  df-rel 5530
This theorem is referenced by: (None)
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