Theorem sucidg 5720

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 2610	. . 3 ⊢ 𝐴 = 𝐴
2	1	olci 405	. 2 ⊢ (𝐴 ∈ 𝐴 ∨ 𝐴 = 𝐴)
3		elsucg 5709	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ suc 𝐴 ↔ (𝐴 ∈ 𝐴 ∨ 𝐴 = 𝐴)))
4	2, 3	mpbiri 247	1 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ suc 𝐴)

Syntax hints:   → wi 4   ∨ wo 382   = wceq 1475   ∈ wcel 1977  suc csuc 5642

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-un 3545  df-sn 4126  df-suc 5646

This theorem is referenced by:  sucid  5721  nsuceq0  5722  trsuc  5727  sucssel  5736  ordsuc  6906  onpsssuc  6911  nlimsucg  6934  tfrlem11  7371  tfrlem13  7373  tz7.44-2  7390  omeulem1  7549  oeordi  7554  oeeulem  7568  php4  8032  wofib  8333  suc11reg  8399  cantnfle  8451  cantnflt2  8453  cantnfp1lem3  8460  cantnflem1  8469  dfac12lem1  8848  dfac12lem2  8849  ttukeylem3  9216  ttukeylem7  9220  r1wunlim  9438  ontgval  31600  sucneqond  32389  finxpreclem4  32407  finxpsuclem  32410