Theorem sucssel 6429

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 6415	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ suc 𝐴)
2		ssel 3940	. 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 2109   ⊆ wss 3914  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-ss 3931  df-sn 4590  df-suc 6338

This theorem is referenced by:  suc11  6441  ordelsuc  7795  ordsucelsuc  7797  oaordi  8510  nnaordi  8582  unbnn2  9244  ackbij1b  10191  ackbij2  10195  cflm  10203  isf32lem2  10307  indpi  10860  dfon2lem3  35773