MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ordtr3 Structured version   Visualization version   GIF version

Theorem ordtr3 5986
Description: Transitive law for ordinal classes. (Contributed by Mario Carneiro, 30-Dec-2014.) (Proof shortened by JJ, 24-Sep-2021.)
Assertion
Ref Expression
ordtr3 ((Ord 𝐵 ∧ Ord 𝐶) → (𝐴𝐵 → (𝐴𝐶𝐶𝐵)))

Proof of Theorem ordtr3
StepHypRef Expression
1 nelss 3860 . . . . . 6 ((𝐴𝐵 ∧ ¬ 𝐴𝐶) → ¬ 𝐵𝐶)
21adantl 474 . . . . 5 (((Ord 𝐵 ∧ Ord 𝐶) ∧ (𝐴𝐵 ∧ ¬ 𝐴𝐶)) → ¬ 𝐵𝐶)
3 ordtri1 5974 . . . . . . 7 ((Ord 𝐵 ∧ Ord 𝐶) → (𝐵𝐶 ↔ ¬ 𝐶𝐵))
43con2bid 346 . . . . . 6 ((Ord 𝐵 ∧ Ord 𝐶) → (𝐶𝐵 ↔ ¬ 𝐵𝐶))
54adantr 473 . . . . 5 (((Ord 𝐵 ∧ Ord 𝐶) ∧ (𝐴𝐵 ∧ ¬ 𝐴𝐶)) → (𝐶𝐵 ↔ ¬ 𝐵𝐶))
62, 5mpbird 249 . . . 4 (((Ord 𝐵 ∧ Ord 𝐶) ∧ (𝐴𝐵 ∧ ¬ 𝐴𝐶)) → 𝐶𝐵)
76expr 449 . . 3 (((Ord 𝐵 ∧ Ord 𝐶) ∧ 𝐴𝐵) → (¬ 𝐴𝐶𝐶𝐵))
87orrd 890 . 2 (((Ord 𝐵 ∧ Ord 𝐶) ∧ 𝐴𝐵) → (𝐴𝐶𝐶𝐵))
98ex 402 1 ((Ord 𝐵 ∧ Ord 𝐶) → (𝐴𝐵 → (𝐴𝐶𝐶𝐵)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 385  wo 874  wcel 2157  wss 3769  Ord word 5940
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-tr 4946  df-eprel 5225  df-po 5233  df-so 5234  df-fr 5271  df-we 5273  df-ord 5944
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator