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Theorem po2ne 5562
Description: Two sets related by a partial order are not equal. (Contributed by AV, 13-Mar-2023.)
Assertion
Ref Expression
po2ne ((𝑅 Po 𝑉 ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝑅𝐵) → 𝐴𝐵)

Proof of Theorem po2ne
StepHypRef Expression
1 breq1 5109 . . . . 5 (𝐴 = 𝐵 → (𝐴𝑅𝐵𝐵𝑅𝐵))
2 poirr 5558 . . . . . . . . 9 ((𝑅 Po 𝑉𝐵𝑉) → ¬ 𝐵𝑅𝐵)
32adantrl 715 . . . . . . . 8 ((𝑅 Po 𝑉 ∧ (𝐴𝑉𝐵𝑉)) → ¬ 𝐵𝑅𝐵)
43pm2.21d 121 . . . . . . 7 ((𝑅 Po 𝑉 ∧ (𝐴𝑉𝐵𝑉)) → (𝐵𝑅𝐵𝐴𝐵))
54ex 414 . . . . . 6 (𝑅 Po 𝑉 → ((𝐴𝑉𝐵𝑉) → (𝐵𝑅𝐵𝐴𝐵)))
65com13 88 . . . . 5 (𝐵𝑅𝐵 → ((𝐴𝑉𝐵𝑉) → (𝑅 Po 𝑉𝐴𝐵)))
71, 6syl6bi 253 . . . 4 (𝐴 = 𝐵 → (𝐴𝑅𝐵 → ((𝐴𝑉𝐵𝑉) → (𝑅 Po 𝑉𝐴𝐵))))
87com24 95 . . 3 (𝐴 = 𝐵 → (𝑅 Po 𝑉 → ((𝐴𝑉𝐵𝑉) → (𝐴𝑅𝐵𝐴𝐵))))
983impd 1349 . 2 (𝐴 = 𝐵 → ((𝑅 Po 𝑉 ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝑅𝐵) → 𝐴𝐵))
10 ax-1 6 . 2 (𝐴𝐵 → ((𝑅 Po 𝑉 ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝑅𝐵) → 𝐴𝐵))
119, 10pm2.61ine 3029 1 ((𝑅 Po 𝑉 ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝑅𝐵) → 𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 397  w3a 1088   = wceq 1542  wcel 2107  wne 2944   class class class wbr 5106   Po wpo 5544
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 2708
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 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-ral 3066  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-po 5546
This theorem is referenced by:  prproropf1olem1  45702
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