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Theorem onsuci 7779
Description: The successor of an ordinal number is an ordinal number. Inference associated with onsuc 7751 and onsucb 7757. Corollary 7N(c) of [Enderton] p. 193. (Contributed by NM, 12-Jun-1994.)
Hypothesis
Ref Expression
onssi.1 𝐴 ∈ On
Assertion
Ref Expression
onsuci suc 𝐴 ∈ On

Proof of Theorem onsuci
StepHypRef Expression
1 onssi.1 . 2 𝐴 ∈ On
2 onsuc 7751 . 2 (𝐴 ∈ On → suc 𝐴 ∈ On)
31, 2ax-mp 5 1 suc 𝐴 ∈ On
Colors of variables: wff setvar class
Syntax hints:  wcel 2107  Oncon0 6322  suc csuc 6324
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-tr 5228  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-ord 6325  df-on 6326  df-suc 6328
This theorem is referenced by:  1onOLD  8430  2onOLD  8432  3on  8435  4on  8436  tz9.12lem2  9731  tz9.12  9733  rankpwi  9766  bndrank  9784  rankval4  9810  rankmapu  9821  rankxplim3  9824  cfcof  10217  ttukeylem6  10457  onsucconni  34938  onsucsuccmpi  34944
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