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Theorem ordsucsssuc 7823
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 7822 . . . 4 (Ord 𝐴 → (𝐵𝐴 ↔ suc 𝐵 ∈ suc 𝐴))
21notbid 317 . . 3 (Ord 𝐴 → (¬ 𝐵𝐴 ↔ ¬ suc 𝐵 ∈ suc 𝐴))
32adantr 479 . 2 ((Ord 𝐴 ∧ Ord 𝐵) → (¬ 𝐵𝐴 ↔ ¬ suc 𝐵 ∈ suc 𝐴))
4 ordtri1 6397 . 2 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵 ↔ ¬ 𝐵𝐴))
5 ordsuc 7813 . . 3 (Ord 𝐴 ↔ Ord suc 𝐴)
6 ordsuc 7813 . . 3 (Ord 𝐵 ↔ Ord suc 𝐵)
7 ordtri1 6397 . . 3 ((Ord suc 𝐴 ∧ Ord suc 𝐵) → (suc 𝐴 ⊆ suc 𝐵 ↔ ¬ suc 𝐵 ∈ suc 𝐴))
85, 6, 7syl2anb 596 . 2 ((Ord 𝐴 ∧ Ord 𝐵) → (suc 𝐴 ⊆ suc 𝐵 ↔ ¬ suc 𝐵 ∈ suc 𝐴))
93, 4, 83bitr4d 310 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 394  wcel 2098  wss 3940  Ord word 6363  suc csuc 6366
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 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423  ax-un 7737
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3960  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-tr 5261  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-ord 6367  df-on 6368  df-suc 6370
This theorem is referenced by:  oawordri  8567  oeworde  8610  nnawordi  8638  eldifsucnn  8681  ttrcltr  9737  bndrank  9862  rankmapu  9899  ackbij1b  10260  onsuct0  35981  finxpsuclem  36932  onsucwordi  42781  naddgeoa  42888
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