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Theorem po2ne 5487
Description: Two classes which are in a partial order relation 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 5065 . . . . 5 (𝐴 = 𝐵 → (𝐴𝑅𝐵𝐵𝑅𝐵))
2 poirr 5483 . . . . . . . . 9 ((𝑅 Po 𝑉𝐵𝑉) → ¬ 𝐵𝑅𝐵)
32adantrl 712 . . . . . . . 8 ((𝑅 Po 𝑉 ∧ (𝐴𝑉𝐵𝑉)) → ¬ 𝐵𝑅𝐵)
43pm2.21d 121 . . . . . . 7 ((𝑅 Po 𝑉 ∧ (𝐴𝑉𝐵𝑉)) → (𝐵𝑅𝐵𝐴𝐵))
54ex 413 . . . . . 6 (𝑅 Po 𝑉 → ((𝐴𝑉𝐵𝑉) → (𝐵𝑅𝐵𝐴𝐵)))
65com13 88 . . . . 5 (𝐵𝑅𝐵 → ((𝐴𝑉𝐵𝑉) → (𝑅 Po 𝑉𝐴𝐵)))
71, 6syl6bi 254 . . . 4 (𝐴 = 𝐵 → (𝐴𝑅𝐵 → ((𝐴𝑉𝐵𝑉) → (𝑅 Po 𝑉𝐴𝐵))))
87com24 95 . . 3 (𝐴 = 𝐵 → (𝑅 Po 𝑉 → ((𝐴𝑉𝐵𝑉) → (𝐴𝑅𝐵𝐴𝐵))))
983impd 1342 . 2 (𝐴 = 𝐵 → ((𝑅 Po 𝑉 ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝑅𝐵) → 𝐴𝐵))
10 ax-1 6 . 2 (𝐴𝐵 → ((𝑅 Po 𝑉 ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝑅𝐵) → 𝐴𝐵))
119, 10pm2.61ine 3104 1 ((𝑅 Po 𝑉 ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝑅𝐵) → 𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  w3a 1081   = wceq 1530  wcel 2106  wne 3020   class class class wbr 5062   Po wpo 5470
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rab 3151  df-v 3501  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-br 5063  df-po 5472
This theorem is referenced by:  prproropf1olem1  43494
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