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Theorem snex 3934
Description: A singleton whose element exists is a set. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 24-May-2019.)
Hypothesis
Ref Expression
snex.1 𝐴 ∈ V
Assertion
Ref Expression
snex {𝐴} ∈ V

Proof of Theorem snex
StepHypRef Expression
1 snex.1 . 2 𝐴 ∈ V
2 snexg 3933 . 2 (𝐴 ∈ V → {𝐴} ∈ V)
31, 2ax-mp 7 1 {𝐴} ∈ V
Colors of variables: wff set class
Syntax hints:  wcel 1393  Vcvv 2554  {csn 3372
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3872  ax-pow 3924
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2556  df-in 2921  df-ss 2928  df-pw 3358  df-sn 3378
This theorem is referenced by:  ensn1  6239  xpsnen  6258  endisj  6261  xpcomco  6263  xpassen  6267  phplem2  6279  findcard2  6308  findcard2s  6309  ac6sfi  6314  nn0ex  8135
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