Theorem onsuci 7784

Description: The successor of an ordinal number is an ordinal number. Inference associated with onsuc 7758 and onsucb 7762. 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 7758	. 2 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ On)
3	1, 2	ax-mp 5	1 ⊢ suc 𝐴 ∈ On

Syntax hints:   ∈ wcel 2114  Oncon0 6318  suc csuc 6320

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5232  ax-pr 5371  ax-un 7683

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-tr 5194  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-ord 6321  df-on 6322  df-suc 6324

This theorem is referenced by:  3on  8415  4on  8416  tz9.12lem2  9706  tz9.12  9708  rankpwi  9741  bndrank  9759  rankval4  9785  rankmapu  9796  rankxplim3  9799  cfcof  10190  ttukeylem6  10430  bdayiun  27924  n0bday  28361  bdaypw2n0bndlem  28472  bdaypw2bnd  28474  bdayfinbndlem1  28476  z12bdaylem2  28480  rankval4b  35262  onsucconni  36638  onsucsuccmpi  36644