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Theorem gruwun 8451
Description: A nonempty Grothendieck's universe is a weak universe. (Contributed by Mario Carneiro, 2-Jan-2017.)
Assertion
Ref Expression
gruwun  |-  ( ( U  e.  Univ  /\  U  =/=  (/) )  ->  U  e. WUni )

Proof of Theorem gruwun
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 grutr 8431 . . 3  |-  ( U  e.  Univ  ->  Tr  U
)
21adantr 451 . 2  |-  ( ( U  e.  Univ  /\  U  =/=  (/) )  ->  Tr  U )
3 simpr 447 . 2  |-  ( ( U  e.  Univ  /\  U  =/=  (/) )  ->  U  =/=  (/) )
4 gruuni 8438 . . . . 5  |-  ( ( U  e.  Univ  /\  x  e.  U )  ->  U. x  e.  U )
54adantlr 695 . . . 4  |-  ( ( ( U  e.  Univ  /\  U  =/=  (/) )  /\  x  e.  U )  ->  U. x  e.  U
)
6 grupw 8433 . . . . 5  |-  ( ( U  e.  Univ  /\  x  e.  U )  ->  ~P x  e.  U )
76adantlr 695 . . . 4  |-  ( ( ( U  e.  Univ  /\  U  =/=  (/) )  /\  x  e.  U )  ->  ~P x  e.  U
)
8 grupr 8435 . . . . . . 7  |-  ( ( U  e.  Univ  /\  x  e.  U  /\  y  e.  U )  ->  { x ,  y }  e.  U )
983expa 1151 . . . . . 6  |-  ( ( ( U  e.  Univ  /\  x  e.  U )  /\  y  e.  U
)  ->  { x ,  y }  e.  U )
109adantllr 699 . . . . 5  |-  ( ( ( ( U  e. 
Univ  /\  U  =/=  (/) )  /\  x  e.  U )  /\  y  e.  U
)  ->  { x ,  y }  e.  U )
1110ralrimiva 2639 . . . 4  |-  ( ( ( U  e.  Univ  /\  U  =/=  (/) )  /\  x  e.  U )  ->  A. y  e.  U  { x ,  y }  e.  U )
125, 7, 113jca 1132 . . 3  |-  ( ( ( U  e.  Univ  /\  U  =/=  (/) )  /\  x  e.  U )  ->  ( U. x  e.  U  /\  ~P x  e.  U  /\  A. y  e.  U  { x ,  y }  e.  U ) )
1312ralrimiva 2639 . 2  |-  ( ( U  e.  Univ  /\  U  =/=  (/) )  ->  A. x  e.  U  ( U. x  e.  U  /\  ~P x  e.  U  /\  A. y  e.  U  { x ,  y }  e.  U ) )
14 iswun 8342 . . 3  |-  ( U  e.  Univ  ->  ( U  e. WUni 
<->  ( Tr  U  /\  U  =/=  (/)  /\  A. x  e.  U  ( U. x  e.  U  /\  ~P x  e.  U  /\  A. y  e.  U  { x ,  y }  e.  U ) ) ) )
1514adantr 451 . 2  |-  ( ( U  e.  Univ  /\  U  =/=  (/) )  ->  ( U  e. WUni  <->  ( Tr  U  /\  U  =/=  (/)  /\  A. x  e.  U  ( U. x  e.  U  /\  ~P x  e.  U  /\  A. y  e.  U  { x ,  y }  e.  U ) ) ) )
162, 3, 13, 15mpbir3and 1135 1  |-  ( ( U  e.  Univ  /\  U  =/=  (/) )  ->  U  e. WUni )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    e. wcel 1696    =/= wne 2459   A.wral 2556   (/)c0 3468   ~Pcpw 3638   {cpr 3654   U.cuni 3843   Tr wtr 4129  WUnicwun 8338   Univcgru 8428
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-map 6790  df-wun 8340  df-gru 8429
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