Theorem onsuci 4643

Description: The successor of an ordinal number is an ordinal number. Inference associated with onsuc 4628 and onsucb 4630. Corollary 7N(c) of [Enderton] p. 193. (Contributed by NM, 12-Jun-1994.)

Hypothesis

Ref	Expression
onssi.1	|- A e. On

Assertion

Ref	Expression
onsuci	|- suc A e. On

Proof of Theorem onsuci

Step	Hyp	Ref	Expression
1		onssi.1	. 2 |- A e. On
2		onsuc 4628	. 2 |- ( A e. On -> suc A e. On )
3	1, 2	ax-mp 5	1 |- suc A e. On

Syntax hints:   e. wcel 2205   Oncon0 4489   suc csuc 4491

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-uni 3920  df-tr 4214  df-iord 4492  df-on 4494  df-suc 4497

This theorem is referenced by:  ordtri2orexmid  4650  onsucsssucexmid  4654  ordsoexmid  4689  ordtri2or2exmid  4698  ontri2orexmidim  4699  tfr0dm  6566  1on  6667  2on  6669  3on  6671  4on  6673  onntri35  7560  onntri45  7564  prarloclemarch2  7750