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Theorem tskr1om 8405
Description: A nonempty Tarski's class is infinite, because it contains all the finite levels of the cumulative hierarchy. (This proof does not use ax-inf 7355.) (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 7455 . . . . . 6  |-  R1  Fn  On
2 fnfun 5357 . . . . . 6  |-  ( R1  Fn  On  ->  Fun  R1 )
31, 2ax-mp 8 . . . . 5  |-  Fun  R1
4 fvelima 5590 . . . . 5  |-  ( ( Fun  R1  /\  y  e.  ( R1 " om ) )  ->  E. x  e.  om  ( R1 `  x )  =  y )
53, 4mpan 651 . . . 4  |-  ( y  e.  ( R1 " om )  ->  E. x  e.  om  ( R1 `  x )  =  y )
6 fveq2 5541 . . . . . . . 8  |-  ( x  =  (/)  ->  ( R1
`  x )  =  ( R1 `  (/) ) )
76eleq1d 2362 . . . . . . 7  |-  ( x  =  (/)  ->  ( ( R1 `  x )  e.  T  <->  ( R1 `  (/) )  e.  T
) )
8 fveq2 5541 . . . . . . . 8  |-  ( x  =  y  ->  ( R1 `  x )  =  ( R1 `  y
) )
98eleq1d 2362 . . . . . . 7  |-  ( x  =  y  ->  (
( R1 `  x
)  e.  T  <->  ( R1 `  y )  e.  T
) )
10 fveq2 5541 . . . . . . . 8  |-  ( x  =  suc  y  -> 
( R1 `  x
)  =  ( R1
`  suc  y )
)
1110eleq1d 2362 . . . . . . 7  |-  ( x  =  suc  y  -> 
( ( R1 `  x )  e.  T  <->  ( R1 `  suc  y
)  e.  T ) )
12 r10 7456 . . . . . . . 8  |-  ( R1
`  (/) )  =  (/)
13 tsk0 8401 . . . . . . . 8  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  (/)  e.  T
)
1412, 13syl5eqel 2380 . . . . . . 7  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  ( R1 `  (/) )  e.  T
)
15 tskpw 8391 . . . . . . . . . 10  |-  ( ( T  e.  Tarski  /\  ( R1 `  y )  e.  T )  ->  ~P ( R1 `  y )  e.  T )
16 nnon 4678 . . . . . . . . . . . 12  |-  ( y  e.  om  ->  y  e.  On )
17 r1suc 7458 . . . . . . . . . . . 12  |-  ( y  e.  On  ->  ( R1 `  suc  y )  =  ~P ( R1
`  y ) )
1816, 17syl 15 . . . . . . . . . . 11  |-  ( y  e.  om  ->  ( R1 `  suc  y )  =  ~P ( R1
`  y ) )
1918eleq1d 2362 . . . . . . . . . 10  |-  ( y  e.  om  ->  (
( R1 `  suc  y )  e.  T  <->  ~P ( R1 `  y
)  e.  T ) )
2015, 19syl5ibr 212 . . . . . . . . 9  |-  ( y  e.  om  ->  (
( T  e.  Tarski  /\  ( R1 `  y
)  e.  T )  ->  ( R1 `  suc  y )  e.  T
) )
2120exp3a 425 . . . . . . . 8  |-  ( y  e.  om  ->  ( T  e.  Tarski  ->  (
( R1 `  y
)  e.  T  -> 
( R1 `  suc  y )  e.  T
) ) )
2221adantrd 454 . . . . . . 7  |-  ( y  e.  om  ->  (
( T  e.  Tarski  /\  T  =/=  (/) )  -> 
( ( R1 `  y )  e.  T  ->  ( R1 `  suc  y )  e.  T
) ) )
237, 9, 11, 14, 22finds2 4700 . . . . . 6  |-  ( x  e.  om  ->  (
( T  e.  Tarski  /\  T  =/=  (/) )  -> 
( R1 `  x
)  e.  T ) )
24 eleq1 2356 . . . . . . 7  |-  ( ( R1 `  x )  =  y  ->  (
( R1 `  x
)  e.  T  <->  y  e.  T ) )
2524imbi2d 307 . . . . . 6  |-  ( ( R1 `  x )  =  y  ->  (
( ( T  e. 
Tarski  /\  T  =/=  (/) )  -> 
( R1 `  x
)  e.  T )  <-> 
( ( T  e. 
Tarski  /\  T  =/=  (/) )  -> 
y  e.  T ) ) )
2623, 25syl5ibcom 211 . . . . 5  |-  ( x  e.  om  ->  (
( R1 `  x
)  =  y  -> 
( ( T  e. 
Tarski  /\  T  =/=  (/) )  -> 
y  e.  T ) ) )
2726rexlimiv 2674 . . . 4  |-  ( E. x  e.  om  ( R1 `  x )  =  y  ->  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  y  e.  T ) )
285, 27syl 15 . . 3  |-  ( y  e.  ( R1 " om )  ->  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  y  e.  T ) )
2928com12 27 . 2  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  (
y  e.  ( R1
" om )  -> 
y  e.  T ) )
3029ssrdv 3198 1  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  ( R1 " om )  C_  T )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   E.wrex 2557    C_ wss 3165   (/)c0 3468   ~Pcpw 3638   Oncon0 4408   suc csuc 4410   omcom 4672   "cima 4708   Fun wfun 5265    Fn wfn 5266   ` cfv 5271   R1cr1 7450   Tarskictsk 8386
This theorem is referenced by:  tskr1om2  8406  tskinf  8407
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
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-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-iun 3923  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-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-recs 6404  df-rdg 6439  df-r1 7452  df-tsk 8387
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