Theorem onsuci 4553

Description: The successor of an ordinal number is an ordinal number. Inference associated with onsuc 4538 and onsucb 4540. 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 4538	. 2 |- ( A e. On -> suc A e. On )
3	1, 2	ax-mp 5	1 |- suc A e. On

Syntax hints:   e. wcel 2167   Oncon0 4399   suc csuc 4401

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-uni 3841  df-tr 4133  df-iord 4402  df-on 4404  df-suc 4407

This theorem is referenced by:  ordtri2orexmid  4560  onsucsssucexmid  4564  ordsoexmid  4599  ordtri2or2exmid  4608  ontri2orexmidim  4609  tfr0dm  6389  1on  6490  2on  6492  3on  6494  4on  6495  onntri35  7320  onntri45  7324  prarloclemarch2  7503