Theorem sucex 6903

Description: The successor of a set is a set. (Contributed by NM, 30-Aug-1993.)

Hypothesis

Ref	Expression
sucex.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
sucex	⊢ suc 𝐴 ∈ V

Proof of Theorem sucex

Step	Hyp	Ref	Expression
1		sucex.1	. 2 ⊢ 𝐴 ∈ V
2		sucexg 6902	. 2 ⊢ (𝐴 ∈ V → suc 𝐴 ∈ V)
3	1, 2	ax-mp 5	1 ⊢ suc 𝐴 ∈ V

Syntax hints:   ∈ wcel 1977  Vcvv 3173  suc csuc 5642

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-sn 4126  df-pr 4128  df-uni 4373  df-suc 5646

This theorem is referenced by:  orduninsuc  6935  tfindsg  6952  tfinds2  6955  finds  6984  findsg  6985  finds2  6986  seqomlem1  7432  oasuc  7491  onasuc  7495  infensuc  8023  phplem4  8027  php  8029  inf0  8401  inf3lem1  8408  dfom3  8427  cantnflt  8452  cantnflem1  8469  cnfcom  8480  infxpenlem  8719  pwsdompw  8909  ackbij1lem5  8929  cfslb2n  8973  cfsmolem  8975  fin1a2lem12  9116  axdc4lem  9160  alephreg  9283  bnj986  30278  bnj1018  30286  dfon2lem7  30938  bj-1ex  32131  bj-2ex  32132  dford3lem2  36612