Theorem elelsuc 6407

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 867	. 2 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵))
2		elsucg 6402	. 2 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ suc 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)))
3	1, 2	mpbird 257	1 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ suc 𝐵)

Syntax hints:   → wi 4   ∨ wo 847   = wceq 1540   ∈ wcel 2109  suc csuc 6334

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-v 3449  df-un 3919  df-sn 4590  df-suc 6338

This theorem is referenced by:  suctr  6420  pssnn  9132  ttrcltr  9669  ttrclss  9673  ttrclselem2  9679  pwsdompw  10156  fin1a2lem4  10356  grur1a  10772  bnj570  34895  satom  35343  satfv0  35345  satfvsuc  35348  satf00  35361  satf0suc  35363  sat1el2xp  35366  fmla  35368  fmla0  35369  fmlasuc0  35371  satfdmfmla  35387  finxpsuclem  37385