Theorem onsuci 4608

Description: The successor of an ordinal number is an ordinal number. Inference associated with onsuc 4593 and onsucb 4595. 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 4593	. 2 |- ( A e. On -> suc A e. On )
3	1, 2	ax-mp 5	1 |- suc A e. On

Syntax hints:   e. wcel 2200   Oncon0 4454   suc csuc 4456

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-uni 3889  df-tr 4183  df-iord 4457  df-on 4459  df-suc 4462

This theorem is referenced by:  ordtri2orexmid  4615  onsucsssucexmid  4619  ordsoexmid  4654  ordtri2or2exmid  4663  ontri2orexmidim  4664  tfr0dm  6468  1on  6569  2on  6571  3on  6573  4on  6574  onntri35  7422  onntri45  7426  prarloclemarch2  7606