Theorem onsuc 4593

Description: The successor of an ordinal number is an ordinal number. Closed form of onsuci 4608. Forward implication of onsucb 4595. 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 4466	. . 3 |- ( A e. On -> Ord A )
2		ordsucim 4592	. . 3 |- ( Ord A -> Ord suc A )
3	1, 2	syl 14	. 2 |- ( A e. On -> Ord suc A )
4		sucexg 4590	. . 3 |- ( A e. On -> suc A e. _V )
5		elong 4464	. . 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 2799   Ord word 4453   Oncon0 4454   suc csuc 4456

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 4202  ax-pow 4258  ax-pr 4293  ax-un 4524

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 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-uni 3889  df-tr 4183  df-iord 4457  df-on 4459  df-suc 4462

This theorem is referenced by:  onsucb  4595  unon  4603  onsuci  4608  ordsucunielexmid  4623  tfrlemisucaccv  6477  tfrexlem  6486  tfri1dALT  6503  rdgisuc1  6536  rdgon  6538  oacl  6614  oasuc  6618  omsuc  6626