Theorem sucidg 4559

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 2238	. . 3 ⊢ 𝐴 = 𝐴
2	1	olci 744	. 2 ⊢ (𝐴 ∈ 𝐴 ∨ 𝐴 = 𝐴)
3		elsucg 4547	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ suc 𝐴 ↔ (𝐴 ∈ 𝐴 ∨ 𝐴 = 𝐴)))
4	2, 3	mpbiri 168	1 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ suc 𝐴)

Syntax hints:   → wi 4   ∨ wo 720   = wceq 1402   ∈ wcel 2209  suc csuc 4508

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-sn 3714  df-suc 4514

This theorem is referenced by:  sucid  4560  nsuceq0g  4561  trsuc  4565  sucssel  4567  ordsucg  4647  sucunielr  4655  suc11g  4702  nlimsucg  4711  peano2b  4760  omsinds  4767  nnpredlt  4769  frecsuclem  6671  phplem4dom  7157  phplem4on  7163  dif1en  7177  fin0  7183  fin0or  7184  fidcenumlemrks  7264  bj-peano4  16964