Theorem sucssel 6343

Description: A set whose successor is a subset of another class is a member of that class. (Contributed by NM, 16-Sep-1995.)

Assertion

Ref	Expression
sucssel	⊢ (𝐴 ∈ 𝑉 → (suc 𝐴 ⊆ 𝐵 → 𝐴 ∈ 𝐵))

Proof of Theorem sucssel

Step	Hyp	Ref	Expression
1		sucidg 6329	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ suc 𝐴)
2		ssel 3910	. 2 ⊢ (suc 𝐴 ⊆ 𝐵 → (𝐴 ∈ suc 𝐴 → 𝐴 ∈ 𝐵))
3	1, 2	syl5com 31	1 ⊢ (𝐴 ∈ 𝑉 → (suc 𝐴 ⊆ 𝐵 → 𝐴 ∈ 𝐵))

Syntax hints:   → wi 4   ∈ wcel 2108   ⊆ wss 3883  suc csuc 6253

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-v 3424  df-un 3888  df-in 3890  df-ss 3900  df-sn 4559  df-suc 6257

This theorem is referenced by:  suc11  6354  ordelsuc  7642  ordsucelsuc  7644  oaordi  8339  nnaordi  8411  unbnn2  9001  ackbij1b  9926  ackbij2  9930  cflm  9937  isf32lem2  10041  indpi  10594  dfon2lem3  33667