Theorem sucexg 7745

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 3459	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		sucexb 7744	. 2 ⊢ (𝐴 ∈ V ↔ suc 𝐴 ∈ V)
3	1, 2	sylib 218	1 ⊢ (𝐴 ∈ 𝑉 → suc 𝐴 ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2109  Vcvv 3438  suc csuc 6313

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 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374  ax-un 7675

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-sn 4580  df-pr 4582  df-uni 4862  df-suc 6317

This theorem is referenced by:  sucex  7746  onsuc  7751  cofon1  8597  cofon2  8598  hsmexlem1  10339  dfon2lem3  35778  inaex  44290