Theorem onsuci 7884

Description: The successor of an ordinal number is an ordinal number. Inference associated with onsuc 7858 and onsucb 7862. 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

Step	Hyp	Ref	Expression
1		onssi.1	. 2 ⊢ 𝐴 ∈ On
2		onsuc 7858	. 2 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ On)
3	1, 2	ax-mp 5	1 ⊢ suc 𝐴 ∈ On

This proof depends on syntax axioms:   ∈ wcel 2147  Oncon0 6388  suc csuc 6390

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5255  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-tr 5219  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-ord 6391  df-on 6392  df-suc 6394

This theorem is used by:  3on  8531  4on  8532  tz9.12lem2  9832  tz9.12  9834  rankpwi  9869  bndrank  9891  rankval4b  9917  rankval4  9921  rankmapu  9932  rankxplim3  9935  cfcof  10391  ttukeylem6  10631  bdayiun  28358  n0bday  28795  bdaypw2n0bndlem  28906  bdaypw2bnd  28908  bdayfinbndlem1  28910  z12bdaylem2  28914  scottssr1  35881  5on  35904  6on  35905  7on  35906  8on  35907  9on  35908  onprcf1acwevdlem2  36087  onsucconni  37440  onsucsuccmpi  37446