Theorem onsuc 4648

Description: The successor of an ordinal number is an ordinal number. Closed form of onsuci 4663. Forward implication of onsucb 4650. Proposition 7.24 of [TakeutiZaring] p. 41. (Contributed by NM, 6-Jun-1994.)

Assertion

Ref	Expression
onsuc	|- ( A e. On -> suc A e. On )

Proof of Theorem onsuc

Step	Hyp	Ref	Expression
1		eloni 4520	. . 3 |- ( A e. On -> Ord A )
2		ordsucim 4647	. . 3 |- ( Ord A -> Ord suc A )
3	1, 2	syl 14	. 2 |- ( A e. On -> Ord suc A )
4		sucexg 4645	. . 3 |- ( A e. On -> suc A e. _V )
5		elong 4518	. . 3 |- ( suc A e. _V -> ( suc A e. On <-> Ord suc A ) )
6	4, 5	syl 14	. 2 |- ( A e. On -> ( suc A e. On <-> Ord suc A ) )
7	3, 6	mpbird 167	1 |- ( A e. On -> suc A e. On )

This proof depends on syntax axioms:   -> wi 4   <-> wb 105   e. wcel 2209   _Vcvv 2821   Ord word 4507   Oncon0 4508   suc csuc 4510

This proof depends on 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 4249  ax-pow 4311  ax-pr 4346  ax-un 4578

This proof 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 3690  df-sn 3715  df-pr 3716  df-uni 3936  df-tr 4230  df-iord 4511  df-on 4513  df-suc 4516

This theorem is used by:  onsucb  4650  unon  4658  onsuci  4663  ordsucunielexmid  4678  tfrlemisucaccv  6596  tfrexlem  6605  tfri1dALT  6622  rdgisuc1  6655  rdgon  6657  oacl  6733  oasuc  6737  omsuc  6745