Theorem onsuci 7831

Description: The successor of an ordinal number is an ordinal number. Inference associated with onsuc 7805 and onsucb 7809. 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 7805	. 2 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ On)
3	1, 2	ax-mp 5	1 ⊢ suc 𝐴 ∈ On

Syntax hints:   ∈ wcel 2143  Oncon0 6360  suc csuc 6362

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5257  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-tr 5219  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-ord 6363  df-on 6364  df-suc 6366

This theorem is referenced by:  3on  8466  4on  8467  tz9.12lem2  9756  tz9.12  9758  rankpwi  9791  bndrank  9809  rankval4  9835  rankmapu  9846  rankxplim3  9849  cfcof  10253  ttukeylem6  10493  bdayiun  28108  n0bday  28545  bdaypw2n0bndlem  28656  bdaypw2bnd  28658  bdayfinbndlem1  28660  z12bdaylem2  28664  rankval4b  35493  scottssr1  35524  onsucconni  36948  onsucsuccmpi  36954