Theorem onsuci 7794

Description: The successor of an ordinal number is an ordinal number. Inference associated with onsuc 7767 and onsucb 7772. 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 7767	. 2 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ On)
3	1, 2	ax-mp 5	1 ⊢ suc 𝐴 ∈ On

Syntax hints:   ∈ wcel 2109  Oncon0 6320  suc csuc 6322

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-tr 5210  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-ord 6323  df-on 6324  df-suc 6326

This theorem is referenced by:  3on  8427  4on  8428  tz9.12lem2  9717  tz9.12  9719  rankpwi  9752  bndrank  9770  rankval4  9796  rankmapu  9807  rankxplim3  9810  cfcof  10203  ttukeylem6  10443  bdayiun  27864  n0sbday  28284  onsucconni  36418  onsucsuccmpi  36424