Theorem sucexg 7825

Description: The successor of a set is a set (generalization). (Contributed by NM, 5-Jun-1994.)

Assertion

Ref	Expression
sucexg	⊢ (𝐴 ∈ 𝑉 → suc 𝐴 ∈ V)

Proof of Theorem sucexg

Step	Hyp	Ref	Expression
1		elex 3501	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		sucexb 7824	. 2 ⊢ (𝐴 ∈ V ↔ suc 𝐴 ∈ V)
3	1, 2	sylib 218	1 ⊢ (𝐴 ∈ 𝑉 → suc 𝐴 ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2108  Vcvv 3480  suc csuc 6386

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 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-sn 4627  df-pr 4629  df-uni 4908  df-suc 6390

This theorem is referenced by:  sucex  7826  onsuc  7831  suceloniOLD  7832  cofon1  8710  cofon2  8711  hsmexlem1  10466  dfon2lem3  35786  inaex  44316