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Theorem gruwun 8680
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 8660 . . 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 8667 . . . . 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 8662 . . . . 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 8664 . . . . . . 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 2781 . . . 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 2781 . 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 8571 . . 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 1725    =/= wne 2598   A.wral 2697   (/)c0 3620   ~Pcpw 3791   {cpr 3807   U.cuni 4007   Tr wtr 4294  WUnicwun 8567   Univcgru 8657
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-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
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-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-map 7012  df-wun 8569  df-gru 8658
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