Theorem sucssel 6460

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 6446	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ suc 𝐴)
2		ssel 3902	. 2 ⊢ (suc 𝐴 ⊆ 𝐵 → (𝐴 ∈ suc 𝐴 → 𝐴 ∈ 𝐵))
3	1, 2	syl5com 32	1 ⊢ (𝐴 ∈ 𝑉 → (suc 𝐴 ⊆ 𝐵 → 𝐴 ∈ 𝐵))

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2144   ⊆ wss 3895  suc csuc 6364

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-ss 3896  df-un 4151  df-sn 4585  df-suc 6368

This theorem is used by:  suc11  6472  ordelsuc  7832  ordsucelsuc  7834  oaordi  8558  nnaordi  8631  unbnn2  9293  ackbij1b  10320  ackbij2  10324  cflm  10331  isf32lem2  10436  indpi  10996  dfon2lem3  36640