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Theorem grusn 8643
Description: A Grothendieck's universe contains the singletons of its elements. (Contributed by Mario Carneiro, 9-Jun-2013.)
Assertion
Ref Expression
grusn  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  { A }  e.  U )

Proof of Theorem grusn
StepHypRef Expression
1 dfsn2 3796 . 2  |-  { A }  =  { A ,  A }
2 grupr 8636 . . 3  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  A  e.  U )  ->  { A ,  A }  e.  U
)
323anidm23 1243 . 2  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  { A ,  A }  e.  U
)
41, 3syl5eqel 2496 1  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  { A }  e.  U )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    e. wcel 1721   {csn 3782   {cpr 3783   Univcgru 8629
This theorem is referenced by:  gruop  8644
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-tr 4271  df-iota 5385  df-fv 5429  df-ov 6051  df-gru 8630
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