Theorem elelsuc 5714

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 399	. 2 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵))
2		elsucg 5709	. 2 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ suc 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)))
3	1, 2	mpbird 246	1 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ suc 𝐵)

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:  suctr  5725  suctrOLD  5726  pssnn  8063  pwsdompw  8909  fin1a2lem4  9108  grur1a  9520  bnj570  30229  finxpsuclem  32410