Theorem sucidg 4102

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 ∈ 𝑉 → A ∈ suc A)

Proof of Theorem sucidg

Step	Hyp	Ref	Expression
1		eqid 2022	. . 3 ⊢ A = A
2	1	olci 638	. 2 ⊢ (A ∈ A ∨ A = A)
3		elsucg 4090	. 2 ⊢ (A ∈ 𝑉 → (A ∈ suc A ↔ (A ∈ A ∨ A = A)))
4	2, 3	mpbiri 157	1 ⊢ (A ∈ 𝑉 → A ∈ suc A)

Syntax hints:   → wi 4   ∨ wo 616   = wceq 1228   ∈ wcel 1374  suc csuc 4051

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004

This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-un 2899  df-sn 3356  df-suc 4057

This theorem is referenced by:  sucid  4103  nsuceq0g  4104  trsuc  4109  sucssel  4111  ordsucg  4178  sucunielr  4185  suc11g  4219  nlimsucg  4226  onpsssuc  4231  peano2b  4264  frecsuclem2  5905  bj-peano4  7177