Theorem sucexg 7797

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 3492	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		sucexb 7796	. 2 ⊢ (𝐴 ∈ V ↔ suc 𝐴 ∈ V)
3	1, 2	sylib 217	1 ⊢ (𝐴 ∈ 𝑉 → suc 𝐴 ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2105  Vcvv 3473  suc csuc 6366

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7729

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-sn 4629  df-pr 4631  df-uni 4909  df-suc 6370

This theorem is referenced by:  sucex  7798  onsuc  7803  suceloniOLD  7804  cofon1  8677  cofon2  8678  hsmexlem1  10427  dfon2lem3  35228  inaex  43521