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Theorem onsucss 43256
Description: If one ordinal is less than another, then the successor of the first is less than or equal to the second. Lemma 1.13 of [Schloeder] p. 2. See ordsucss 7838. (Contributed by RP, 16-Jan-2025.)
Assertion
Ref Expression
onsucss (𝐴 ∈ On → (𝐵𝐴 → suc 𝐵𝐴))

Proof of Theorem onsucss
StepHypRef Expression
1 eloni 6396 . 2 (𝐴 ∈ On → Ord 𝐴)
2 ordsucss 7838 . 2 (Ord 𝐴 → (𝐵𝐴 → suc 𝐵𝐴))
31, 2syl 17 1 (𝐴 ∈ On → (𝐵𝐴 → suc 𝐵𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106  wss 3963  Ord word 6385  Oncon0 6386  suc csuc 6388
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-ord 6389  df-on 6390  df-suc 6392
This theorem is referenced by:  oege2  43297  cantnftermord  43310  tfsconcatb0  43334  naddwordnexlem0  43386  naddwordnexlem4  43391
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