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Theorem sucexg 7825
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 3499 . 2 (𝐴𝑉𝐴 ∈ V)
2 sucexb 7824 . 2 (𝐴 ∈ V ↔ suc 𝐴 ∈ V)
31, 2sylib 218 1 (𝐴𝑉 → suc 𝐴 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106  Vcvv 3478  suc csuc 6388
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-sn 4632  df-pr 4634  df-uni 4913  df-suc 6392
This theorem is referenced by:  sucex  7826  onsuc  7831  suceloniOLD  7832  cofon1  8709  cofon2  8710  hsmexlem1  10464  dfon2lem3  35767  inaex  44293
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