Theorem sucssel 6398

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 6384	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ suc 𝐴)
2		ssel 3923	. 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 2111   ⊆ wss 3897  suc csuc 6303

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-v 3438  df-un 3902  df-ss 3914  df-sn 4572  df-suc 6307

This theorem is referenced by:  suc11  6410  ordelsuc  7745  ordsucelsuc  7747  oaordi  8456  nnaordi  8528  unbnn2  9176  ackbij1b  10124  ackbij2  10128  cflm  10136  isf32lem2  10240  indpi  10793  dfon2lem3  35819