Theorem sucexg 7752

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 3454	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		sucexb 7751	. 2 ⊢ (𝐴 ∈ V ↔ suc 𝐴 ∈ V)
3	1, 2	sylib 220	1 ⊢ (𝐴 ∈ 𝑉 → suc 𝐴 ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2121  Vcvv 3433  suc csuc 6316

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713  ax-sep 5221  ax-pr 5365  ax-un 7682

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-rab 3394  df-v 3435  df-un 3890  df-in 3892  df-ss 3902  df-sn 4559  df-pr 4561  df-uni 4842  df-suc 6320

This theorem is referenced by:  sucex  7753  onsuc  7757  cofon1  8602  cofon2  8603  hsmexlem1  10343  fineqvnttrclse  35320  dfon2lem3  36026  dmsucmap  38850  sucmapsuc  38871  inaex  44756