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Theorem ordtr3 5930
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 3805 . . . . . 6 ((𝐴𝐵 ∧ ¬ 𝐴𝐶) → ¬ 𝐵𝐶)
21adantl 473 . . . . 5 (((Ord 𝐵 ∧ Ord 𝐶) ∧ (𝐴𝐵 ∧ ¬ 𝐴𝐶)) → ¬ 𝐵𝐶)
3 ordtri1 5917 . . . . . . 7 ((Ord 𝐵 ∧ Ord 𝐶) → (𝐵𝐶 ↔ ¬ 𝐶𝐵))
43con2bid 343 . . . . . 6 ((Ord 𝐵 ∧ Ord 𝐶) → (𝐶𝐵 ↔ ¬ 𝐵𝐶))
54adantr 472 . . . . 5 (((Ord 𝐵 ∧ Ord 𝐶) ∧ (𝐴𝐵 ∧ ¬ 𝐴𝐶)) → (𝐶𝐵 ↔ ¬ 𝐵𝐶))
62, 5mpbird 247 . . . 4 (((Ord 𝐵 ∧ Ord 𝐶) ∧ (𝐴𝐵 ∧ ¬ 𝐴𝐶)) → 𝐶𝐵)
76expr 644 . . 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 196  wo 382  wa 383  wcel 2139  wss 3715  Ord word 5883
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-tr 4905  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-ord 5887
This theorem is referenced by: (None)
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