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Theorem tskr1om 9191
Description: A nonempty Tarski class is infinite, because it contains all the finite levels of the cumulative hierarchy. (This proof does not use ax-inf 8143.) (Contributed by Mario Carneiro, 24-Jun-2013.)
Assertion
Ref Expression
tskr1om  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  ( R1 " om )  C_  T )

Proof of Theorem tskr1om
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 r1fnon 8237 . . . . . 6  |-  R1  Fn  On
2 fnfun 5691 . . . . . 6  |-  ( R1  Fn  On  ->  Fun  R1 )
31, 2ax-mp 5 . . . . 5  |-  Fun  R1
4 fvelima 5933 . . . . 5  |-  ( ( Fun  R1  /\  y  e.  ( R1 " om ) )  ->  E. x  e.  om  ( R1 `  x )  =  y )
53, 4mpan 674 . . . 4  |-  ( y  e.  ( R1 " om )  ->  E. x  e.  om  ( R1 `  x )  =  y )
6 fveq2 5881 . . . . . . . 8  |-  ( x  =  (/)  ->  ( R1
`  x )  =  ( R1 `  (/) ) )
76eleq1d 2498 . . . . . . 7  |-  ( x  =  (/)  ->  ( ( R1 `  x )  e.  T  <->  ( R1 `  (/) )  e.  T
) )
8 fveq2 5881 . . . . . . . 8  |-  ( x  =  y  ->  ( R1 `  x )  =  ( R1 `  y
) )
98eleq1d 2498 . . . . . . 7  |-  ( x  =  y  ->  (
( R1 `  x
)  e.  T  <->  ( R1 `  y )  e.  T
) )
10 fveq2 5881 . . . . . . . 8  |-  ( x  =  suc  y  -> 
( R1 `  x
)  =  ( R1
`  suc  y )
)
1110eleq1d 2498 . . . . . . 7  |-  ( x  =  suc  y  -> 
( ( R1 `  x )  e.  T  <->  ( R1 `  suc  y
)  e.  T ) )
12 r10 8238 . . . . . . . 8  |-  ( R1
`  (/) )  =  (/)
13 tsk0 9187 . . . . . . . 8  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  (/)  e.  T
)
1412, 13syl5eqel 2521 . . . . . . 7  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  ( R1 `  (/) )  e.  T
)
15 tskpw 9177 . . . . . . . . . 10  |-  ( ( T  e.  Tarski  /\  ( R1 `  y )  e.  T )  ->  ~P ( R1 `  y )  e.  T )
16 nnon 6712 . . . . . . . . . . . 12  |-  ( y  e.  om  ->  y  e.  On )
17 r1suc 8240 . . . . . . . . . . . 12  |-  ( y  e.  On  ->  ( R1 `  suc  y )  =  ~P ( R1
`  y ) )
1816, 17syl 17 . . . . . . . . . . 11  |-  ( y  e.  om  ->  ( R1 `  suc  y )  =  ~P ( R1
`  y ) )
1918eleq1d 2498 . . . . . . . . . 10  |-  ( y  e.  om  ->  (
( R1 `  suc  y )  e.  T  <->  ~P ( R1 `  y
)  e.  T ) )
2015, 19syl5ibr 224 . . . . . . . . 9  |-  ( y  e.  om  ->  (
( T  e.  Tarski  /\  ( R1 `  y
)  e.  T )  ->  ( R1 `  suc  y )  e.  T
) )
2120expd 437 . . . . . . . 8  |-  ( y  e.  om  ->  ( T  e.  Tarski  ->  (
( R1 `  y
)  e.  T  -> 
( R1 `  suc  y )  e.  T
) ) )
2221adantrd 469 . . . . . . 7  |-  ( y  e.  om  ->  (
( T  e.  Tarski  /\  T  =/=  (/) )  -> 
( ( R1 `  y )  e.  T  ->  ( R1 `  suc  y )  e.  T
) ) )
237, 9, 11, 14, 22finds2 6735 . . . . . 6  |-  ( x  e.  om  ->  (
( T  e.  Tarski  /\  T  =/=  (/) )  -> 
( R1 `  x
)  e.  T ) )
24 eleq1 2501 . . . . . . 7  |-  ( ( R1 `  x )  =  y  ->  (
( R1 `  x
)  e.  T  <->  y  e.  T ) )
2524imbi2d 317 . . . . . 6  |-  ( ( R1 `  x )  =  y  ->  (
( ( T  e. 
Tarski  /\  T  =/=  (/) )  -> 
( R1 `  x
)  e.  T )  <-> 
( ( T  e. 
Tarski  /\  T  =/=  (/) )  -> 
y  e.  T ) ) )
2623, 25syl5ibcom 223 . . . . 5  |-  ( x  e.  om  ->  (
( R1 `  x
)  =  y  -> 
( ( T  e. 
Tarski  /\  T  =/=  (/) )  -> 
y  e.  T ) ) )
2726rexlimiv 2918 . . . 4  |-  ( E. x  e.  om  ( R1 `  x )  =  y  ->  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  y  e.  T ) )
285, 27syl 17 . . 3  |-  ( y  e.  ( R1 " om )  ->  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  y  e.  T ) )
2928com12 32 . 2  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  (
y  e.  ( R1
" om )  -> 
y  e.  T ) )
3029ssrdv 3476 1  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  ( R1 " om )  C_  T )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1870    =/= wne 2625   E.wrex 2783    C_ wss 3442   (/)c0 3767   ~Pcpw 3985   "cima 4857   Oncon0 5442   suc csuc 5444   Fun wfun 5595    Fn wfn 5596   ` cfv 5601   omcom 6706   R1cr1 8232   Tarskictsk 9172
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-om 6707  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-r1 8234  df-tsk 9173
This theorem is referenced by:  tskr1om2  9192  tskinf  9193
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