Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  po2ne Structured version   Visualization version   GIF version

Theorem po2ne 5461
 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 5038 . . . . 5 (𝐴 = 𝐵 → (𝐴𝑅𝐵𝐵𝑅𝐵))
2 poirr 5457 . . . . . . . . 9 ((𝑅 Po 𝑉𝐵𝑉) → ¬ 𝐵𝑅𝐵)
32adantrl 715 . . . . . . . 8 ((𝑅 Po 𝑉 ∧ (𝐴𝑉𝐵𝑉)) → ¬ 𝐵𝑅𝐵)
43pm2.21d 121 . . . . . . 7 ((𝑅 Po 𝑉 ∧ (𝐴𝑉𝐵𝑉)) → (𝐵𝑅𝐵𝐴𝐵))
54ex 416 . . . . . 6 (𝑅 Po 𝑉 → ((𝐴𝑉𝐵𝑉) → (𝐵𝑅𝐵𝐴𝐵)))
65com13 88 . . . . 5 (𝐵𝑅𝐵 → ((𝐴𝑉𝐵𝑉) → (𝑅 Po 𝑉𝐴𝐵)))
71, 6syl6bi 256 . . . 4 (𝐴 = 𝐵 → (𝐴𝑅𝐵 → ((𝐴𝑉𝐵𝑉) → (𝑅 Po 𝑉𝐴𝐵))))
87com24 95 . . 3 (𝐴 = 𝐵 → (𝑅 Po 𝑉 → ((𝐴𝑉𝐵𝑉) → (𝐴𝑅𝐵𝐴𝐵))))
983impd 1345 . 2 (𝐴 = 𝐵 → ((𝑅 Po 𝑉 ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝑅𝐵) → 𝐴𝐵))
10 ax-1 6 . 2 (𝐴𝐵 → ((𝑅 Po 𝑉 ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝑅𝐵) → 𝐴𝐵))
119, 10pm2.61ine 3034 1 ((𝑅 Po 𝑉 ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝑅𝐵) → 𝐴𝐵)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ≠ wne 2951   class class class wbr 5035   Po wpo 5444 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 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729 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 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-v 3411  df-un 3865  df-sn 4526  df-pr 4528  df-op 4532  df-br 5036  df-po 5446 This theorem is referenced by:  prproropf1olem1  44416
 Copyright terms: Public domain W3C validator