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Theorem sucexg 7503
 Description: The successor of a set is a set (generalization). (Contributed by NM, 5-Jun-1994.)
Assertion
Ref Expression
sucexg (𝐴𝑉 → suc 𝐴 ∈ V)

Proof of Theorem sucexg
StepHypRef Expression
1 elex 3491 . 2 (𝐴𝑉𝐴 ∈ V)
2 sucexb 7502 . 2 (𝐴 ∈ V ↔ suc 𝐴 ∈ V)
31, 2sylib 220 1 (𝐴𝑉 → suc 𝐴 ∈ V)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2114  Vcvv 3473  suc csuc 6169 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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5179  ax-nul 5186  ax-pr 5306  ax-un 7439 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-rab 3134  df-v 3475  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-sn 4544  df-pr 4546  df-uni 4815  df-suc 6173 This theorem is referenced by:  sucex  7504  suceloni  7506  hsmexlem1  9826  dfon2lem3  33038  inaex  40788
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