Theorem elelsuc 6338

Description: Membership in a successor. (Contributed by NM, 20-Jun-1998.)

Assertion

Ref	Expression
elelsuc	⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ suc 𝐵)

Proof of Theorem elelsuc

Step	Hyp	Ref	Expression
1		orc 864	. 2 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵))
2		elsucg 6333	. 2 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ suc 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)))
3	1, 2	mpbird 256	1 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ suc 𝐵)

Syntax hints:   → wi 4   ∨ wo 844   = wceq 1539   ∈ wcel 2106  suc csuc 6268

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3434  df-un 3892  df-sn 4562  df-suc 6272

This theorem is referenced by:  suctr  6349  pssnn  8951  pssnnOLD  9040  ttrcltr  9474  ttrclss  9478  ttrclselem2  9484  pwsdompw  9960  fin1a2lem4  10159  grur1a  10575  bnj570  32885  satom  33318  satfv0  33320  satfvsuc  33323  satf00  33336  satf0suc  33338  sat1el2xp  33341  fmla  33343  fmla0  33344  fmlasuc0  33346  satfdmfmla  33362  finxpsuclem  35568