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Theorem ordtr3 6429
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 4049 . . . . . 6 ((𝐴𝐵 ∧ ¬ 𝐴𝐶) → ¬ 𝐵𝐶)
21adantl 481 . . . . 5 (((Ord 𝐵 ∧ Ord 𝐶) ∧ (𝐴𝐵 ∧ ¬ 𝐴𝐶)) → ¬ 𝐵𝐶)
3 ordtri1 6417 . . . . . . 7 ((Ord 𝐵 ∧ Ord 𝐶) → (𝐵𝐶 ↔ ¬ 𝐶𝐵))
43con2bid 354 . . . . . 6 ((Ord 𝐵 ∧ Ord 𝐶) → (𝐶𝐵 ↔ ¬ 𝐵𝐶))
54adantr 480 . . . . 5 (((Ord 𝐵 ∧ Ord 𝐶) ∧ (𝐴𝐵 ∧ ¬ 𝐴𝐶)) → (𝐶𝐵 ↔ ¬ 𝐵𝐶))
62, 5mpbird 257 . . . 4 (((Ord 𝐵 ∧ Ord 𝐶) ∧ (𝐴𝐵 ∧ ¬ 𝐴𝐶)) → 𝐶𝐵)
76expr 456 . . 3 (((Ord 𝐵 ∧ Ord 𝐶) ∧ 𝐴𝐵) → (¬ 𝐴𝐶𝐶𝐵))
87orrd 864 . 2 (((Ord 𝐵 ∧ Ord 𝐶) ∧ 𝐴𝐵) → (𝐴𝐶𝐶𝐵))
98ex 412 1 ((Ord 𝐵 ∧ Ord 𝐶) → (𝐴𝐵 → (𝐴𝐶𝐶𝐵)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 848  wcel 2108  wss 3951  Ord word 6383
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-tr 5260  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-ord 6387
This theorem is referenced by: (None)
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