Theorem sucidg 4276

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 2100	. . 3 ⊢ 𝐴 = 𝐴
2	1	olci 692	. 2 ⊢ (𝐴 ∈ 𝐴 ∨ 𝐴 = 𝐴)
3		elsucg 4264	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ suc 𝐴 ↔ (𝐴 ∈ 𝐴 ∨ 𝐴 = 𝐴)))
4	2, 3	mpbiri 167	1 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ suc 𝐴)

Syntax hints:   → wi 4   ∨ wo 670   = wceq 1299   ∈ wcel 1448  suc csuc 4225

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082

This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-v 2643  df-un 3025  df-sn 3480  df-suc 4231

This theorem is referenced by:  sucid  4277  nsuceq0g  4278  trsuc  4282  sucssel  4284  ordsucg  4356  sucunielr  4364  suc11g  4410  nlimsucg  4419  peano2b  4466  omsinds  4473  frecsuclem  6233  phplem4dom  6685  phplem4on  6690  dif1en  6702  fin0  6708  fin0or  6709  fidcenumlemrks  6769  bj-peano4  12738