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Theorem gruwun 9086
Description: A nonempty Grothendieck 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 9066 . . 3  |-  ( U  e.  Univ  ->  Tr  U
)
21adantr 465 . 2  |-  ( ( U  e.  Univ  /\  U  =/=  (/) )  ->  Tr  U )
3 simpr 461 . 2  |-  ( ( U  e.  Univ  /\  U  =/=  (/) )  ->  U  =/=  (/) )
4 gruuni 9073 . . . . 5  |-  ( ( U  e.  Univ  /\  x  e.  U )  ->  U. x  e.  U )
54adantlr 714 . . . 4  |-  ( ( ( U  e.  Univ  /\  U  =/=  (/) )  /\  x  e.  U )  ->  U. x  e.  U
)
6 grupw 9068 . . . . 5  |-  ( ( U  e.  Univ  /\  x  e.  U )  ->  ~P x  e.  U )
76adantlr 714 . . . 4  |-  ( ( ( U  e.  Univ  /\  U  =/=  (/) )  /\  x  e.  U )  ->  ~P x  e.  U
)
8 grupr 9070 . . . . . . 7  |-  ( ( U  e.  Univ  /\  x  e.  U  /\  y  e.  U )  ->  { x ,  y }  e.  U )
983expa 1188 . . . . . 6  |-  ( ( ( U  e.  Univ  /\  x  e.  U )  /\  y  e.  U
)  ->  { x ,  y }  e.  U )
109adantllr 718 . . . . 5  |-  ( ( ( ( U  e. 
Univ  /\  U  =/=  (/) )  /\  x  e.  U )  /\  y  e.  U
)  ->  { x ,  y }  e.  U )
1110ralrimiva 2827 . . . 4  |-  ( ( ( U  e.  Univ  /\  U  =/=  (/) )  /\  x  e.  U )  ->  A. y  e.  U  { x ,  y }  e.  U )
125, 7, 113jca 1168 . . 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 2827 . 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 8977 . . 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 465 . 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 1171 1  |-  ( ( U  e.  Univ  /\  U  =/=  (/) )  ->  U  e. WUni )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    e. wcel 1758    =/= wne 2645   A.wral 2796   (/)c0 3740   ~Pcpw 3963   {cpr 3982   U.cuni 4194   Tr wtr 4488  WUnicwun 8973   Univcgru 9063
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-sbc 3289  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-fv 5529  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-map 7321  df-wun 8975  df-gru 9064
This theorem is referenced by: (None)
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