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Theorem grutsk1 8443
Description: Grothendieck's universes are the same as transitive Tarski's classes, part one: a transitive Tarski class is a universe. (The hard work is in tskuni 8405.) (Contributed by Mario Carneiro, 17-Jun-2013.)
Assertion
Ref Expression
grutsk1  |-  ( ( T  e.  Tarski  /\  Tr  T )  ->  T  e.  Univ )

Proof of Theorem grutsk1
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 447 . 2  |-  ( ( T  e.  Tarski  /\  Tr  T )  ->  Tr  T )
2 tskpw 8375 . . . . 5  |-  ( ( T  e.  Tarski  /\  x  e.  T )  ->  ~P x  e.  T )
32adantlr 695 . . . 4  |-  ( ( ( T  e.  Tarski  /\ 
Tr  T )  /\  x  e.  T )  ->  ~P x  e.  T
)
4 tskpr 8392 . . . . . . 7  |-  ( ( T  e.  Tarski  /\  x  e.  T  /\  y  e.  T )  ->  { x ,  y }  e.  T )
543expa 1151 . . . . . 6  |-  ( ( ( T  e.  Tarski  /\  x  e.  T )  /\  y  e.  T
)  ->  { x ,  y }  e.  T )
65ralrimiva 2626 . . . . 5  |-  ( ( T  e.  Tarski  /\  x  e.  T )  ->  A. y  e.  T  { x ,  y }  e.  T )
76adantlr 695 . . . 4  |-  ( ( ( T  e.  Tarski  /\ 
Tr  T )  /\  x  e.  T )  ->  A. y  e.  T  { x ,  y }  e.  T )
8 elmapg 6785 . . . . . . 7  |-  ( ( T  e.  Tarski  /\  x  e.  T )  ->  (
y  e.  ( T  ^m  x )  <->  y :
x --> T ) )
98adantlr 695 . . . . . 6  |-  ( ( ( T  e.  Tarski  /\ 
Tr  T )  /\  x  e.  T )  ->  ( y  e.  ( T  ^m  x )  <-> 
y : x --> T ) )
10 tskurn 8411 . . . . . . 7  |-  ( ( ( T  e.  Tarski  /\ 
Tr  T )  /\  x  e.  T  /\  y : x --> T )  ->  U. ran  y  e.  T )
11103expia 1153 . . . . . 6  |-  ( ( ( T  e.  Tarski  /\ 
Tr  T )  /\  x  e.  T )  ->  ( y : x --> T  ->  U. ran  y  e.  T ) )
129, 11sylbid 206 . . . . 5  |-  ( ( ( T  e.  Tarski  /\ 
Tr  T )  /\  x  e.  T )  ->  ( y  e.  ( T  ^m  x )  ->  U. ran  y  e.  T ) )
1312ralrimiv 2625 . . . 4  |-  ( ( ( T  e.  Tarski  /\ 
Tr  T )  /\  x  e.  T )  ->  A. y  e.  ( T  ^m  x ) U. ran  y  e.  T )
143, 7, 133jca 1132 . . 3  |-  ( ( ( T  e.  Tarski  /\ 
Tr  T )  /\  x  e.  T )  ->  ( ~P x  e.  T  /\  A. y  e.  T  { x ,  y }  e.  T  /\  A. y  e.  ( T  ^m  x
) U. ran  y  e.  T ) )
1514ralrimiva 2626 . 2  |-  ( ( T  e.  Tarski  /\  Tr  T )  ->  A. x  e.  T  ( ~P x  e.  T  /\  A. y  e.  T  {
x ,  y }  e.  T  /\  A. y  e.  ( T  ^m  x ) U. ran  y  e.  T )
)
16 elgrug 8414 . . 3  |-  ( T  e.  Tarski  ->  ( T  e. 
Univ 
<->  ( Tr  T  /\  A. x  e.  T  ( ~P x  e.  T  /\  A. y  e.  T  { x ,  y }  e.  T  /\  A. y  e.  ( T  ^m  x ) U. ran  y  e.  T
) ) ) )
1716adantr 451 . 2  |-  ( ( T  e.  Tarski  /\  Tr  T )  ->  ( T  e.  Univ  <->  ( Tr  T  /\  A. x  e.  T  ( ~P x  e.  T  /\  A. y  e.  T  { x ,  y }  e.  T  /\  A. y  e.  ( T  ^m  x
) U. ran  y  e.  T ) ) ) )
181, 15, 17mpbir2and 888 1  |-  ( ( T  e.  Tarski  /\  Tr  T )  ->  T  e.  Univ )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    e. wcel 1684   A.wral 2543   ~Pcpw 3625   {cpr 3641   U.cuni 3827   Tr wtr 4113   ran crn 4690   -->wf 5251  (class class class)co 5858    ^m cmap 6772   Tarskictsk 8370   Univcgru 8412
This theorem is referenced by:  grutsk  8444
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-ac2 8089
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-smo 6363  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-oi 7225  df-har 7272  df-r1 7436  df-card 7572  df-aleph 7573  df-cf 7574  df-acn 7575  df-ac 7743  df-wina 8306  df-ina 8307  df-tsk 8371  df-gru 8413
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