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

Theorem po2ne 5595
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 5142 . . . . 5 (𝐴 = 𝐵 → (𝐴𝑅𝐵𝐵𝑅𝐵))
2 poirr 5591 . . . . . . . . 9 ((𝑅 Po 𝑉𝐵𝑉) → ¬ 𝐵𝑅𝐵)
32adantrl 713 . . . . . . . 8 ((𝑅 Po 𝑉 ∧ (𝐴𝑉𝐵𝑉)) → ¬ 𝐵𝑅𝐵)
43pm2.21d 121 . . . . . . 7 ((𝑅 Po 𝑉 ∧ (𝐴𝑉𝐵𝑉)) → (𝐵𝑅𝐵𝐴𝐵))
54ex 412 . . . . . 6 (𝑅 Po 𝑉 → ((𝐴𝑉𝐵𝑉) → (𝐵𝑅𝐵𝐴𝐵)))
65com13 88 . . . . 5 (𝐵𝑅𝐵 → ((𝐴𝑉𝐵𝑉) → (𝑅 Po 𝑉𝐴𝐵)))
71, 6biimtrdi 252 . . . 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 3017 1 ((𝑅 Po 𝑉 ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝑅𝐵) → 𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  w3a 1084   = wceq 1533  wcel 2098  wne 2932   class class class wbr 5139   Po wpo 5577
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-ral 3054  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-br 5140  df-po 5579
This theorem is referenced by:  prproropf1olem1  46717
  Copyright terms: Public domain W3C validator