Theorem onsuci 4658

Description: The successor of an ordinal number is an ordinal number. Inference associated with onsuc 4643 and onsucb 4645. 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 4643	. 2 |- ( A e. On -> suc A e. On )
3	1, 2	ax-mp 5	1 |- suc A e. On

Syntax hints:   e. wcel 2209   Oncon0 4503   suc csuc 4505

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-uni 3931  df-tr 4225  df-iord 4506  df-on 4508  df-suc 4511

This theorem is referenced by:  ordtri2orexmid  4665  onsucsssucexmid  4669  ordsoexmid  4704  ordtri2or2exmid  4713  ontri2orexmidim  4714  tfr0dm  6583  1on  6684  2on  6686  3on  6688  4on  6690  onntri35  7586  onntri45  7590  prarloclemarch2  7776