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Theorem ordsucsssuc 6970
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 6969 . . . 4 (Ord 𝐴 → (𝐵𝐴 ↔ suc 𝐵 ∈ suc 𝐴))
21notbid 308 . . 3 (Ord 𝐴 → (¬ 𝐵𝐴 ↔ ¬ suc 𝐵 ∈ suc 𝐴))
32adantr 481 . 2 ((Ord 𝐴 ∧ Ord 𝐵) → (¬ 𝐵𝐴 ↔ ¬ suc 𝐵 ∈ suc 𝐴))
4 ordtri1 5715 . 2 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵 ↔ ¬ 𝐵𝐴))
5 ordsuc 6961 . . 3 (Ord 𝐴 ↔ Ord suc 𝐴)
6 ordsuc 6961 . . 3 (Ord 𝐵 ↔ Ord suc 𝐵)
7 ordtri1 5715 . . 3 ((Ord suc 𝐴 ∧ Ord suc 𝐵) → (suc 𝐴 ⊆ suc 𝐵 ↔ ¬ suc 𝐵 ∈ suc 𝐴))
85, 6, 7syl2anb 496 . 2 ((Ord 𝐴 ∧ Ord 𝐵) → (suc 𝐴 ⊆ suc 𝐵 ↔ ¬ suc 𝐵 ∈ suc 𝐴))
93, 4, 83bitr4d 300 1 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵 ↔ suc 𝐴 ⊆ suc 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  wcel 1987  wss 3555  Ord word 5681  suc csuc 5684
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-tr 4713  df-eprel 4985  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-ord 5685  df-on 5686  df-suc 5688
This theorem is referenced by:  oawordri  7575  oeworde  7618  nnawordi  7646  bndrank  8648  rankmapu  8685  ackbij1b  9005  nosino  31575  onsuct0  32082  finxpsuclem  32866
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