Theorem onsuci 7781

Description: The successor of an ordinal number is an ordinal number. Inference associated with onsuc 7755 and onsucb 7759. 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 7755	. 2 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ On)
3	1, 2	ax-mp 5	1 ⊢ suc 𝐴 ∈ On

Syntax hints:   ∈ wcel 2113  Oncon0 6317  suc csuc 6319

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-tr 5206  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-ord 6320  df-on 6321  df-suc 6323

This theorem is referenced by:  3on  8413  4on  8414  tz9.12lem2  9700  tz9.12  9702  rankpwi  9735  bndrank  9753  rankval4  9779  rankmapu  9790  rankxplim3  9793  cfcof  10184  ttukeylem6  10424  bdayiun  27911  n0bday  28348  bdaypw2n0bndlem  28459  bdaypw2bnd  28461  bdayfinbndlem1  28463  z12bdaylem2  28467  rankval4b  35256  onsucconni  36631  onsucsuccmpi  36637