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Theorem sucexg 7242
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 3398 . 2 (𝐴𝑉𝐴 ∈ V)
2 sucexb 7241 . 2 (𝐴 ∈ V ↔ suc 𝐴 ∈ V)
31, 2sylib 210 1 (𝐴𝑉 → suc 𝐴 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2157  Vcvv 3383  suc csuc 5941
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-sep 4973  ax-nul 4981  ax-pr 5095  ax-un 7181
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-rex 3093  df-v 3385  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-sn 4367  df-pr 4369  df-uni 4627  df-suc 5945
This theorem is referenced by:  sucex  7243  suceloni  7245  hsmexlem1  9534  dfon2lem3  32194
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