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Theorem sucexg 4515
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 2763 . 2 (𝐴𝑉𝐴 ∈ V)
2 sucexb 4514 . 2 (𝐴 ∈ V ↔ suc 𝐴 ∈ V)
31, 2sylib 122 1 (𝐴𝑉 → suc 𝐴 ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 2160  Vcvv 2752  suc csuc 4383
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-uni 3825  df-suc 4389
This theorem is referenced by:  sucex  4516  onsuc  4518  peano2  4612  sucinc2  6472  oav2  6489
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