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Theorem gruwun 9223
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 9203 . . 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 9210 . . . . 5  |-  ( ( U  e.  Univ  /\  x  e.  U )  ->  U. x  e.  U )
54adantlr 715 . . . 4  |-  ( ( ( U  e.  Univ  /\  U  =/=  (/) )  /\  x  e.  U )  ->  U. x  e.  U
)
6 grupw 9205 . . . . 5  |-  ( ( U  e.  Univ  /\  x  e.  U )  ->  ~P x  e.  U )
76adantlr 715 . . . 4  |-  ( ( ( U  e.  Univ  /\  U  =/=  (/) )  /\  x  e.  U )  ->  ~P x  e.  U
)
8 grupr 9207 . . . . . . 7  |-  ( ( U  e.  Univ  /\  x  e.  U  /\  y  e.  U )  ->  { x ,  y }  e.  U )
983expa 1199 . . . . . 6  |-  ( ( ( U  e.  Univ  /\  x  e.  U )  /\  y  e.  U
)  ->  { x ,  y }  e.  U )
109adantllr 719 . . . . 5  |-  ( ( ( ( U  e. 
Univ  /\  U  =/=  (/) )  /\  x  e.  U )  /\  y  e.  U
)  ->  { x ,  y }  e.  U )
1110ralrimiva 2820 . . . 4  |-  ( ( ( U  e.  Univ  /\  U  =/=  (/) )  /\  x  e.  U )  ->  A. y  e.  U  { x ,  y }  e.  U )
125, 7, 113jca 1179 . . 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 2820 . 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 9114 . . 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 1182 1  |-  ( ( U  e.  Univ  /\  U  =/=  (/) )  ->  U  e. WUni )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 186    /\ wa 369    /\ w3a 976    e. wcel 1844    =/= wne 2600   A.wral 2756   (/)c0 3740   ~Pcpw 3957   {cpr 3976   U.cuni 4193   Tr wtr 4491  WUnicwun 9110   Univcgru 9200
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3063  df-sbc 3280  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-tr 4492  df-id 4740  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-fv 5579  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-map 7461  df-wun 9112  df-gru 9201
This theorem is referenced by: (None)
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