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Theorem poprelb 46192
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 1138 . . 3 (((Rel 𝑅𝑅 Po 𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐶𝑅𝐷)) → (𝐴𝑋𝐵𝑋))
2 an3 658 . . . . 5 (((Rel 𝑅𝑅 Po 𝑋) ∧ (𝐴𝑅𝐵𝐶𝑅𝐷)) → (Rel 𝑅𝐶𝑅𝐷))
323adant2 1132 . . . 4 (((Rel 𝑅𝑅 Po 𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐶𝑅𝐷)) → (Rel 𝑅𝐶𝑅𝐷))
4 brrelex12 5729 . . . 4 ((Rel 𝑅𝐶𝑅𝐷) → (𝐶 ∈ V ∧ 𝐷 ∈ V))
53, 4syl 17 . . 3 (((Rel 𝑅𝑅 Po 𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐶𝑅𝐷)) → (𝐶 ∈ V ∧ 𝐷 ∈ V))
6 preq12bg 4855 . . 3 (((𝐴𝑋𝐵𝑋) ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶))))
71, 5, 6syl2anc 585 . 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 5154 . . . . . . . . . . 11 ((𝐵 = 𝐶𝐴 = 𝐷) → (𝐵𝑅𝐴𝐶𝑅𝐷))
109ancoms 460 . . . . . . . . . 10 ((𝐴 = 𝐷𝐵 = 𝐶) → (𝐵𝑅𝐴𝐶𝑅𝐷))
1110bicomd 222 . . . . . . . . 9 ((𝐴 = 𝐷𝐵 = 𝐶) → (𝐶𝑅𝐷𝐵𝑅𝐴))
1211anbi2d 630 . . . . . . . 8 ((𝐴 = 𝐷𝐵 = 𝐶) → ((𝐴𝑅𝐵𝐶𝑅𝐷) ↔ (𝐴𝑅𝐵𝐵𝑅𝐴)))
13 po2nr 5603 . . . . . . . . . . . 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 414 . . . . . . . . 9 ((Rel 𝑅𝑅 Po 𝑋) → ((𝐴𝑋𝐵𝑋) → ((𝐴𝑅𝐵𝐵𝑅𝐴) → (𝐴 = 𝐶𝐵 = 𝐷))))
1716com13 88 . . . . . . . 8 ((𝐴𝑅𝐵𝐵𝑅𝐴) → ((𝐴𝑋𝐵𝑋) → ((Rel 𝑅𝑅 Po 𝑋) → (𝐴 = 𝐶𝐵 = 𝐷))))
1812, 17syl6bi 253 . . . . . . 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 1112 . . . 4 (((Rel 𝑅𝑅 Po 𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐶𝑅𝐷)) → ((𝐴 = 𝐷𝐵 = 𝐶) → (𝐴 = 𝐶𝐵 = 𝐷)))
228, 21jaod 858 . . 3 (((Rel 𝑅𝑅 Po 𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐶𝑅𝐷)) → (((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶)) → (𝐴 = 𝐶𝐵 = 𝐷)))
23 orc 866 . . 3 ((𝐴 = 𝐶𝐵 = 𝐷) → ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶)))
2422, 23impbid1 224 . 2 (((Rel 𝑅𝑅 Po 𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐶𝑅𝐷)) → (((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶)) ↔ (𝐴 = 𝐶𝐵 = 𝐷)))
257, 24bitrd 279 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 205  wa 397  wo 846  w3a 1088   = wceq 1542  wcel 2107  Vcvv 3475  {cpr 4631   class class class wbr 5149   Po wpo 5587  Rel wrel 5682
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-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
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-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  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-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-po 5589  df-xp 5683  df-rel 5684
This theorem is referenced by: (None)
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