Theorem onsuc 4550

Description: The successor of an ordinal number is an ordinal number. Closed form of onsuci 4565. Forward implication of onsucb 4552. 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 4423	. . 3 |- ( A e. On -> Ord A )
2		ordsucim 4549	. . 3 |- ( Ord A -> Ord suc A )
3	1, 2	syl 14	. 2 |- ( A e. On -> Ord suc A )
4		sucexg 4547	. . 3 |- ( A e. On -> suc A e. _V )
5		elong 4421	. . 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 2176   _Vcvv 2772   Ord word 4410   Oncon0 4411   suc csuc 4413

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-pow 4219  ax-pr 4254  ax-un 4481

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-uni 3851  df-tr 4144  df-iord 4414  df-on 4416  df-suc 4419

This theorem is referenced by:  onsucb  4552  unon  4560  onsuci  4565  ordsucunielexmid  4580  tfrlemisucaccv  6413  tfrexlem  6422  tfri1dALT  6439  rdgisuc1  6472  rdgon  6474  oacl  6548  oasuc  6552  omsuc  6560