Theorem sucexg 7010

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 3212	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		sucexb 7009	. 2 ⊢ (𝐴 ∈ V ↔ suc 𝐴 ∈ V)
3	1, 2	sylib 208	1 ⊢ (𝐴 ∈ 𝑉 → suc 𝐴 ∈ V)

Syntax hints:   → wi 4   ∈ wcel 1990  Vcvv 3200  suc csuc 5725

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-sn 4178  df-pr 4180  df-uni 4437  df-suc 5729

This theorem is referenced by:  sucex  7011  suceloni  7013  hsmexlem1  9248  dfon2lem3  31690