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Theorem gruwun 8644
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 8624 . . 3  |-  ( U  e.  Univ  ->  Tr  U
)
21adantr 452 . 2  |-  ( ( U  e.  Univ  /\  U  =/=  (/) )  ->  Tr  U )
3 simpr 448 . 2  |-  ( ( U  e.  Univ  /\  U  =/=  (/) )  ->  U  =/=  (/) )
4 gruuni 8631 . . . . 5  |-  ( ( U  e.  Univ  /\  x  e.  U )  ->  U. x  e.  U )
54adantlr 696 . . . 4  |-  ( ( ( U  e.  Univ  /\  U  =/=  (/) )  /\  x  e.  U )  ->  U. x  e.  U
)
6 grupw 8626 . . . . 5  |-  ( ( U  e.  Univ  /\  x  e.  U )  ->  ~P x  e.  U )
76adantlr 696 . . . 4  |-  ( ( ( U  e.  Univ  /\  U  =/=  (/) )  /\  x  e.  U )  ->  ~P x  e.  U
)
8 grupr 8628 . . . . . . 7  |-  ( ( U  e.  Univ  /\  x  e.  U  /\  y  e.  U )  ->  { x ,  y }  e.  U )
983expa 1153 . . . . . 6  |-  ( ( ( U  e.  Univ  /\  x  e.  U )  /\  y  e.  U
)  ->  { x ,  y }  e.  U )
109adantllr 700 . . . . 5  |-  ( ( ( ( U  e. 
Univ  /\  U  =/=  (/) )  /\  x  e.  U )  /\  y  e.  U
)  ->  { x ,  y }  e.  U )
1110ralrimiva 2749 . . . 4  |-  ( ( ( U  e.  Univ  /\  U  =/=  (/) )  /\  x  e.  U )  ->  A. y  e.  U  { x ,  y }  e.  U )
125, 7, 113jca 1134 . . 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 2749 . 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 8535 . . 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 452 . 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 1137 1  |-  ( ( U  e.  Univ  /\  U  =/=  (/) )  ->  U  e. WUni )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    e. wcel 1721    =/= wne 2567   A.wral 2666   (/)c0 3588   ~Pcpw 3759   {cpr 3775   U.cuni 3975   Tr wtr 4262  WUnicwun 8531   Univcgru 8621
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-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
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-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-map 6979  df-wun 8533  df-gru 8622
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