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Theorem ordtr3 5686
 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 3627 . . . . . 6 ((𝐴𝐵 ∧ ¬ 𝐴𝐶) → ¬ 𝐵𝐶)
21adantl 481 . . . . 5 (((Ord 𝐵 ∧ Ord 𝐶) ∧ (𝐴𝐵 ∧ ¬ 𝐴𝐶)) → ¬ 𝐵𝐶)
3 ordtri1 5673 . . . . . . 7 ((Ord 𝐵 ∧ Ord 𝐶) → (𝐵𝐶 ↔ ¬ 𝐶𝐵))
43con2bid 343 . . . . . 6 ((Ord 𝐵 ∧ Ord 𝐶) → (𝐶𝐵 ↔ ¬ 𝐵𝐶))
54adantr 480 . . . . 5 (((Ord 𝐵 ∧ Ord 𝐶) ∧ (𝐴𝐵 ∧ ¬ 𝐴𝐶)) → (𝐶𝐵 ↔ ¬ 𝐵𝐶))
62, 5mpbird 246 . . . 4 (((Ord 𝐵 ∧ Ord 𝐶) ∧ (𝐴𝐵 ∧ ¬ 𝐴𝐶)) → 𝐶𝐵)
76expr 641 . . 3 (((Ord 𝐵 ∧ Ord 𝐶) ∧ 𝐴𝐵) → (¬ 𝐴𝐶𝐶𝐵))
87orrd 392 . 2 (((Ord 𝐵 ∧ Ord 𝐶) ∧ 𝐴𝐵) → (𝐴𝐶𝐶𝐵))
98ex 449 1 ((Ord 𝐵 ∧ Ord 𝐶) → (𝐴𝐵 → (𝐴𝐶𝐶𝐵)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   ∈ wcel 1977   ⊆ wss 3540  Ord word 5639 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-tr 4681  df-eprel 4949  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-ord 5643 This theorem is referenced by: (None)
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