Theorem onsuc 4595

Description: The successor of an ordinal number is an ordinal number. Closed form of onsuci 4610. Forward implication of onsucb 4597. 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 4468	. . 3 |- ( A e. On -> Ord A )
2		ordsucim 4594	. . 3 |- ( Ord A -> Ord suc A )
3	1, 2	syl 14	. 2 |- ( A e. On -> Ord suc A )
4		sucexg 4592	. . 3 |- ( A e. On -> suc A e. _V )
5		elong 4466	. . 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 )

Syntax hints:   -> wi 4   <-> wb 105   e. wcel 2200   _Vcvv 2800   Ord word 4455   Oncon0 4456   suc csuc 4458

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 4203  ax-pow 4260  ax-pr 4295  ax-un 4526

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 2802  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-uni 3890  df-tr 4184  df-iord 4459  df-on 4461  df-suc 4464

This theorem is referenced by:  onsucb  4597  unon  4605  onsuci  4610  ordsucunielexmid  4625  tfrlemisucaccv  6484  tfrexlem  6493  tfri1dALT  6510  rdgisuc1  6543  rdgon  6545  oacl  6621  oasuc  6625  omsuc  6633