Theorem sucidg 4399

Description: Part of Proposition 7.23 of [TakeutiZaring] p. 41 (generalized). (Contributed by NM, 25-Mar-1995.) (Proof shortened by Scott Fenton, 20-Feb-2012.)

Assertion

Ref	Expression
sucidg	|- ( A e. V -> A e. suc A )

Proof of Theorem sucidg

Step	Hyp	Ref	Expression
1		eqid 2170	. . 3 |- A = A
2	1	olci 727	. 2 |- ( A e. A \/ A = A )
3		elsucg 4387	. 2 |- ( A e. V -> ( A e. suc A <-> ( A e. A \/ A = A ) ) )
4	2, 3	mpbiri 167	1 |- ( A e. V -> A e. suc A )

Syntax hints:   -> wi 4   \/ wo 703   = wceq 1348   e. wcel 2141   suc csuc 4348

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-sn 3587  df-suc 4354

This theorem is referenced by:  sucid  4400  nsuceq0g  4401  trsuc  4405  sucssel  4407  ordsucg  4484  sucunielr  4492  suc11g  4539  nlimsucg  4548  peano2b  4597  omsinds  4604  nnpredlt  4606  frecsuclem  6383  phplem4dom  6838  phplem4on  6843  dif1en  6855  fin0  6861  fin0or  6862  fidcenumlemrks  6928  bj-peano4  13955