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Theorem sucexg 7514
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 3510 . 2 (𝐴𝑉𝐴 ∈ V)
2 sucexb 7513 . 2 (𝐴 ∈ V ↔ suc 𝐴 ∈ V)
31, 2sylib 219 1 (𝐴𝑉 → suc 𝐴 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2105  Vcvv 3492  suc csuc 6186
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-sn 4558  df-pr 4560  df-uni 4831  df-suc 6190
This theorem is referenced by:  sucex  7515  suceloni  7517  hsmexlem1  9836  dfon2lem3  32927  inaex  40510
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