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Theorem gruop 8680
Description: A Grothendieck's universe contains ordered pairs of its elements. (Contributed by Mario Carneiro, 10-Jun-2013.)
Assertion
Ref Expression
gruop  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  B  e.  U )  ->  <. A ,  B >.  e.  U )

Proof of Theorem gruop
StepHypRef Expression
1 dfopg 3982 . . 3  |-  ( ( A  e.  U  /\  B  e.  U )  -> 
<. A ,  B >.  =  { { A } ,  { A ,  B } } )
213adant1 975 . 2  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  B  e.  U )  ->  <. A ,  B >.  =  { { A } ,  { A ,  B } } )
3 simp1 957 . . 3  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  B  e.  U )  ->  U  e.  Univ )
4 grusn 8679 . . . 4  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  { A }  e.  U )
543adant3 977 . . 3  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  B  e.  U )  ->  { A }  e.  U )
6 grupr 8672 . . 3  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  B  e.  U )  ->  { A ,  B }  e.  U
)
7 grupr 8672 . . 3  |-  ( ( U  e.  Univ  /\  { A }  e.  U  /\  { A ,  B }  e.  U )  ->  { { A } ,  { A ,  B } }  e.  U
)
83, 5, 6, 7syl3anc 1184 . 2  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  B  e.  U )  ->  { { A } ,  { A ,  B } }  e.  U )
92, 8eqeltrd 2510 1  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  B  e.  U )  ->  <. A ,  B >.  e.  U )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 936    = wceq 1652    e. wcel 1725   {csn 3814   {cpr 3815   <.cop 3817   Univcgru 8665
This theorem is referenced by:  gruf  8686
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-tr 4303  df-iota 5418  df-fv 5462  df-ov 6084  df-gru 8666
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