Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  snex GIF version

Theorem snex 3937
 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 3936 . 2 (𝐴 ∈ V → {𝐴} ∈ V)
31, 2ax-mp 7 1 {𝐴} ∈ V
 Colors of variables: wff set class Syntax hints:   ∈ wcel 1393  Vcvv 2557  {csn 3375 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 3875  ax-pow 3927 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 2559  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381 This theorem is referenced by:  ensn1  6276  xpsnen  6295  endisj  6298  xpcomco  6300  xpassen  6304  phplem2  6316  findcard2  6346  findcard2s  6347  ac6sfi  6352  nn0ex  8187
 Copyright terms: Public domain W3C validator