Theorem onsuci 4582

Description: The successor of an ordinal number is an ordinal number. Inference associated with onsuc 4567 and onsucb 4569. 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 4567	. 2 |- ( A e. On -> suc A e. On )
3	1, 2	ax-mp 5	1 |- suc A e. On

Syntax hints:   e. wcel 2178   Oncon0 4428   suc csuc 4430

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-uni 3865  df-tr 4159  df-iord 4431  df-on 4433  df-suc 4436

This theorem is referenced by:  ordtri2orexmid  4589  onsucsssucexmid  4593  ordsoexmid  4628  ordtri2or2exmid  4637  ontri2orexmidim  4638  tfr0dm  6431  1on  6532  2on  6534  3on  6536  4on  6537  onntri35  7383  onntri45  7387  prarloclemarch2  7567