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Theorem grutsk1 8459
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 8421.) (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 8391 . . . . 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 8408 . . . . . . 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 2639 . . . . 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 6801 . . . . . . 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 8427 . . . . . . 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 2638 . . . 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 2639 . 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 8430 . . 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 1696   A.wral 2556   ~Pcpw 3638   {cpr 3654   U.cuni 3843   Tr wtr 4129   ran crn 4706   -->wf 5267  (class class class)co 5874    ^m cmap 6788   Tarskictsk 8386   Univcgru 8428
This theorem is referenced by:  grutsk  8460
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-ac2 8105
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-smo 6379  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-oi 7241  df-har 7288  df-r1 7452  df-card 7588  df-aleph 7589  df-cf 7590  df-acn 7591  df-ac 7759  df-wina 8322  df-ina 8323  df-tsk 8387  df-gru 8429
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