Theorem sucidg 5986

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 2765	. . 3 ⊢ 𝐴 = 𝐴
2	1	olci 892	. 2 ⊢ (𝐴 ∈ 𝐴 ∨ 𝐴 = 𝐴)
3		elsucg 5975	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ suc 𝐴 ↔ (𝐴 ∈ 𝐴 ∨ 𝐴 = 𝐴)))
4	2, 3	mpbiri 249	1 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ suc 𝐴)

Syntax hints:   → wi 4   ∨ wo 873   = wceq 1652   ∈ wcel 2155  suc csuc 5910

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-ext 2743

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-v 3352  df-un 3737  df-sn 4335  df-suc 5914

This theorem is referenced by:  sucid  5987  nsuceq0  5988  trsuc  5992  sucssel  6000  ordsuc  7212  onpsssuc  7217  nlimsucg  7240  tfrlem11  7688  tfrlem13  7690  tz7.44-2  7707  omeulem1  7867  oeordi  7872  oeeulem  7886  php4  8354  wofib  8657  suc11reg  8731  cantnfle  8783  cantnflt2  8785  cantnfp1lem3  8792  cantnflem1  8801  dfac12lem1  9218  dfac12lem2  9219  ttukeylem3  9586  ttukeylem7  9590  r1wunlim  9812  noresle  32290  noprefixmo  32292  ontgval  32869  sucneqond  33646  finxpreclem4  33664  finxpsuclem  33667