Theorem sucidg 4448

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 2193	. . 3 |- A = A
2	1	olci 733	. 2 |- ( A e. A \/ A = A )
3		elsucg 4436	. 2 |- ( A e. V -> ( A e. suc A <-> ( A e. A \/ A = A ) ) )
4	2, 3	mpbiri 168	1 |- ( A e. V -> A e. suc A )

Syntax hints:   -> wi 4   \/ wo 709   = wceq 1364   e. wcel 2164   suc csuc 4397

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-un 3158  df-sn 3625  df-suc 4403

This theorem is referenced by:  sucid  4449  nsuceq0g  4450  trsuc  4454  sucssel  4456  ordsucg  4535  sucunielr  4543  suc11g  4590  nlimsucg  4599  peano2b  4648  omsinds  4655  nnpredlt  4657  frecsuclem  6461  phplem4dom  6920  phplem4on  6925  dif1en  6937  fin0  6943  fin0or  6944  fidcenumlemrks  7014  bj-peano4  15517