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Theorem ordsucsssuc 7808
Description: The subclass relationship between two ordinal classes is inherited by their successors. (Contributed by NM, 4-Oct-2003.)
Assertion
Ref Expression
ordsucsssuc ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵 ↔ suc 𝐴 ⊆ suc 𝐵))

Proof of Theorem ordsucsssuc
StepHypRef Expression
1 ordsucelsuc 7807 . . . 4 (Ord 𝐴 → (𝐵𝐴 ↔ suc 𝐵 ∈ suc 𝐴))
21notbid 318 . . 3 (Ord 𝐴 → (¬ 𝐵𝐴 ↔ ¬ suc 𝐵 ∈ suc 𝐴))
32adantr 480 . 2 ((Ord 𝐴 ∧ Ord 𝐵) → (¬ 𝐵𝐴 ↔ ¬ suc 𝐵 ∈ suc 𝐴))
4 ordtri1 6391 . 2 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵 ↔ ¬ 𝐵𝐴))
5 ordsuc 7798 . . 3 (Ord 𝐴 ↔ Ord suc 𝐴)
6 ordsuc 7798 . . 3 (Ord 𝐵 ↔ Ord suc 𝐵)
7 ordtri1 6391 . . 3 ((Ord suc 𝐴 ∧ Ord suc 𝐵) → (suc 𝐴 ⊆ suc 𝐵 ↔ ¬ suc 𝐵 ∈ suc 𝐴))
85, 6, 7syl2anb 597 . 2 ((Ord 𝐴 ∧ Ord 𝐵) → (suc 𝐴 ⊆ suc 𝐵 ↔ ¬ suc 𝐵 ∈ suc 𝐴))
93, 4, 83bitr4d 311 1 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵 ↔ suc 𝐴 ⊆ suc 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 395  wcel 2098  wss 3943  Ord word 6357  suc csuc 6360
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-tr 5259  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-ord 6361  df-on 6362  df-suc 6364
This theorem is referenced by:  oawordri  8551  oeworde  8594  nnawordi  8622  eldifsucnn  8665  ttrcltr  9713  bndrank  9838  rankmapu  9875  ackbij1b  10236  onsuct0  35834  finxpsuclem  36785  onsucwordi  42619  naddgeoa  42726
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