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Theorem grusn 8684
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 3830 . 2  |-  { A }  =  { A ,  A }
2 grupr 8677 . . 3  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  A  e.  U )  ->  { A ,  A }  e.  U
)
323anidm23 1244 . 2  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  { A ,  A }  e.  U
)
41, 3syl5eqel 2522 1  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  { A }  e.  U )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    e. wcel 1726   {csn 3816   {cpr 3817   Univcgru 8670
This theorem is referenced by:  gruop  8685
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-tr 4306  df-iota 5421  df-fv 5465  df-ov 6087  df-gru 8671
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