Theorem sucidg 6264

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 2821	. . 3 ⊢ 𝐴 = 𝐴
2	1	olci 862	. 2 ⊢ (𝐴 ∈ 𝐴 ∨ 𝐴 = 𝐴)
3		elsucg 6253	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ suc 𝐴 ↔ (𝐴 ∈ 𝐴 ∨ 𝐴 = 𝐴)))
4	2, 3	mpbiri 260	1 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ suc 𝐴)

Syntax hints:   → wi 4   ∨ wo 843   = wceq 1533   ∈ wcel 2110  suc csuc 6188

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-v 3497  df-un 3941  df-sn 4562  df-suc 6192

This theorem is referenced by:  sucid  6265  nsuceq0  6266  trsuc  6270  sucssel  6278  ordsuc  7523  onpsssuc  7528  nlimsucg  7551  tfrlem11  8018  tfrlem13  8020  tz7.44-2  8037  omeulem1  8202  oeordi  8207  oeeulem  8221  php4  8698  wofib  9003  suc11reg  9076  cantnfle  9128  cantnflt2  9130  cantnfp1lem3  9137  cantnflem1  9146  dfac12lem1  9563  dfac12lem2  9564  ttukeylem3  9927  ttukeylem7  9931  r1wunlim  10153  fmla  32623  ex-sategoelelomsuc  32668  noresle  33195  noprefixmo  33197  ontgval  33774  sucneqond  34640  finxpreclem4  34669  finxpsuclem  34672