Theorem sucssel 6481

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 6467	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ suc 𝐴)
2		ssel 3989	. 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 2106   ⊆ wss 3963  suc csuc 6388

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-un 3968  df-ss 3980  df-sn 4632  df-suc 6392

This theorem is referenced by:  suc11  6493  ordelsuc  7840  ordsucelsuc  7842  oaordi  8583  nnaordi  8655  unbnn2  9331  ackbij1b  10276  ackbij2  10280  cflm  10288  isf32lem2  10392  indpi  10945  dfon2lem3  35767