Theorem elelsuc 6437

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 880	. 2 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵))
2		elsucg 6432	. 2 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ suc 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)))
3	1, 2	mpbird 260	1 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ suc 𝐵)

Syntax hints:   → wi 4   ∨ wo 860   = wceq 1567   ∈ wcel 2149  suc csuc 6363

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-v 3465  df-un 3918  df-sn 4595  df-suc 6367

This theorem is referenced by:  suctr  6450  pssnn  9152  ttrcltr  9684  ttrclss  9688  ttrclselem2  9694  pwsdompw  10185  fin1a2lem4  10386  grur1a  10803  bnj570  35237  fineqvnttrclselem3  35458  satom  35746  satfv0  35748  satfvsuc  35751  satf00  35764  satf0suc  35766  sat1el2xp  35769  fmla  35771  fmla0  35772  fmlasuc0  35774  satfdmfmla  35790  nmulprop  36580  finxpsuclem  37930