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Theorem ordtr3 6234
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 4034 . . . . . 6 ((𝐴𝐵 ∧ ¬ 𝐴𝐶) → ¬ 𝐵𝐶)
21adantl 482 . . . . 5 (((Ord 𝐵 ∧ Ord 𝐶) ∧ (𝐴𝐵 ∧ ¬ 𝐴𝐶)) → ¬ 𝐵𝐶)
3 ordtri1 6222 . . . . . . 7 ((Ord 𝐵 ∧ Ord 𝐶) → (𝐵𝐶 ↔ ¬ 𝐶𝐵))
43con2bid 356 . . . . . 6 ((Ord 𝐵 ∧ Ord 𝐶) → (𝐶𝐵 ↔ ¬ 𝐵𝐶))
54adantr 481 . . . . 5 (((Ord 𝐵 ∧ Ord 𝐶) ∧ (𝐴𝐵 ∧ ¬ 𝐴𝐶)) → (𝐶𝐵 ↔ ¬ 𝐵𝐶))
62, 5mpbird 258 . . . 4 (((Ord 𝐵 ∧ Ord 𝐶) ∧ (𝐴𝐵 ∧ ¬ 𝐴𝐶)) → 𝐶𝐵)
76expr 457 . . 3 (((Ord 𝐵 ∧ Ord 𝐶) ∧ 𝐴𝐵) → (¬ 𝐴𝐶𝐶𝐵))
87orrd 859 . 2 (((Ord 𝐵 ∧ Ord 𝐶) ∧ 𝐴𝐵) → (𝐴𝐶𝐶𝐵))
98ex 413 1 ((Ord 𝐵 ∧ Ord 𝐶) → (𝐴𝐵 → (𝐴𝐶𝐶𝐵)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wo 843  wcel 2107  wss 3940  Ord word 6188
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pr 5326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-tr 5170  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-ord 6192
This theorem is referenced by: (None)
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