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Theorem onsuci 7545
Description: The successor of an ordinal number is an ordinal number. 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 suceloni 7520 . 2 (𝐴 ∈ On → suc 𝐴 ∈ On)
31, 2ax-mp 5 1 suc 𝐴 ∈ On
Colors of variables: wff setvar class
Syntax hints:  wcel 2108  Oncon0 6184  suc csuc 6186
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1083  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-tr 5164  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-ord 6187  df-on 6188  df-suc 6190
This theorem is referenced by:  1on  8101  2on  8103  3on  8106  4on  8107  tz9.12lem2  9209  tz9.12  9211  rankpwi  9244  bndrank  9262  rankval4  9288  rankmapu  9299  rankxplim3  9302  cfcof  9688  ttukeylem6  9928  onsucconni  33778  onsucsuccmpi  33784
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