Theorem onsuc 4623

Description: The successor of an ordinal number is an ordinal number. Closed form of onsuci 4638. Forward implication of onsucb 4625. 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 4496	. . 3 |- ( A e. On -> Ord A )
2		ordsucim 4622	. . 3 |- ( Ord A -> Ord suc A )
3	1, 2	syl 14	. 2 |- ( A e. On -> Ord suc A )
4		sucexg 4620	. . 3 |- ( A e. On -> suc A e. _V )
5		elong 4494	. . 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 2203   _Vcvv 2813   Ord word 4483   Oncon0 4484   suc csuc 4486

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-uni 3915  df-tr 4209  df-iord 4487  df-on 4489  df-suc 4492

This theorem is referenced by:  onsucb  4625  unon  4633  onsuci  4638  ordsucunielexmid  4653  tfrlemisucaccv  6556  tfrexlem  6565  tfri1dALT  6582  rdgisuc1  6615  rdgon  6617  oacl  6693  oasuc  6697  omsuc  6705