Theorem onsuc 4599

Description: The successor of an ordinal number is an ordinal number. Closed form of onsuci 4614. Forward implication of onsucb 4601. 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 4472	. . 3 |- ( A e. On -> Ord A )
2		ordsucim 4598	. . 3 |- ( Ord A -> Ord suc A )
3	1, 2	syl 14	. 2 |- ( A e. On -> Ord suc A )
4		sucexg 4596	. . 3 |- ( A e. On -> suc A e. _V )
5		elong 4470	. . 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 2202   _Vcvv 2802   Ord word 4459   Oncon0 4460   suc csuc 4462

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-uni 3894  df-tr 4188  df-iord 4463  df-on 4465  df-suc 4468

This theorem is referenced by:  onsucb  4601  unon  4609  onsuci  4614  ordsucunielexmid  4629  tfrlemisucaccv  6490  tfrexlem  6499  tfri1dALT  6516  rdgisuc1  6549  rdgon  6551  oacl  6627  oasuc  6631  omsuc  6639