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Theorem tskr1om 8642
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 7593.) (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 7693 . . . . . 6  |-  R1  Fn  On
2 fnfun 5542 . . . . . 6  |-  ( R1  Fn  On  ->  Fun  R1 )
31, 2ax-mp 8 . . . . 5  |-  Fun  R1
4 fvelima 5778 . . . . 5  |-  ( ( Fun  R1  /\  y  e.  ( R1 " om ) )  ->  E. x  e.  om  ( R1 `  x )  =  y )
53, 4mpan 652 . . . 4  |-  ( y  e.  ( R1 " om )  ->  E. x  e.  om  ( R1 `  x )  =  y )
6 fveq2 5728 . . . . . . . 8  |-  ( x  =  (/)  ->  ( R1
`  x )  =  ( R1 `  (/) ) )
76eleq1d 2502 . . . . . . 7  |-  ( x  =  (/)  ->  ( ( R1 `  x )  e.  T  <->  ( R1 `  (/) )  e.  T
) )
8 fveq2 5728 . . . . . . . 8  |-  ( x  =  y  ->  ( R1 `  x )  =  ( R1 `  y
) )
98eleq1d 2502 . . . . . . 7  |-  ( x  =  y  ->  (
( R1 `  x
)  e.  T  <->  ( R1 `  y )  e.  T
) )
10 fveq2 5728 . . . . . . . 8  |-  ( x  =  suc  y  -> 
( R1 `  x
)  =  ( R1
`  suc  y )
)
1110eleq1d 2502 . . . . . . 7  |-  ( x  =  suc  y  -> 
( ( R1 `  x )  e.  T  <->  ( R1 `  suc  y
)  e.  T ) )
12 r10 7694 . . . . . . . 8  |-  ( R1
`  (/) )  =  (/)
13 tsk0 8638 . . . . . . . 8  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  (/)  e.  T
)
1412, 13syl5eqel 2520 . . . . . . 7  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  ( R1 `  (/) )  e.  T
)
15 tskpw 8628 . . . . . . . . . 10  |-  ( ( T  e.  Tarski  /\  ( R1 `  y )  e.  T )  ->  ~P ( R1 `  y )  e.  T )
16 nnon 4851 . . . . . . . . . . . 12  |-  ( y  e.  om  ->  y  e.  On )
17 r1suc 7696 . . . . . . . . . . . 12  |-  ( y  e.  On  ->  ( R1 `  suc  y )  =  ~P ( R1
`  y ) )
1816, 17syl 16 . . . . . . . . . . 11  |-  ( y  e.  om  ->  ( R1 `  suc  y )  =  ~P ( R1
`  y ) )
1918eleq1d 2502 . . . . . . . . . 10  |-  ( y  e.  om  ->  (
( R1 `  suc  y )  e.  T  <->  ~P ( R1 `  y
)  e.  T ) )
2015, 19syl5ibr 213 . . . . . . . . 9  |-  ( y  e.  om  ->  (
( T  e.  Tarski  /\  ( R1 `  y
)  e.  T )  ->  ( R1 `  suc  y )  e.  T
) )
2120exp3a 426 . . . . . . . 8  |-  ( y  e.  om  ->  ( T  e.  Tarski  ->  (
( R1 `  y
)  e.  T  -> 
( R1 `  suc  y )  e.  T
) ) )
2221adantrd 455 . . . . . . 7  |-  ( y  e.  om  ->  (
( T  e.  Tarski  /\  T  =/=  (/) )  -> 
( ( R1 `  y )  e.  T  ->  ( R1 `  suc  y )  e.  T
) ) )
237, 9, 11, 14, 22finds2 4873 . . . . . 6  |-  ( x  e.  om  ->  (
( T  e.  Tarski  /\  T  =/=  (/) )  -> 
( R1 `  x
)  e.  T ) )
24 eleq1 2496 . . . . . . 7  |-  ( ( R1 `  x )  =  y  ->  (
( R1 `  x
)  e.  T  <->  y  e.  T ) )
2524imbi2d 308 . . . . . 6  |-  ( ( R1 `  x )  =  y  ->  (
( ( T  e. 
Tarski  /\  T  =/=  (/) )  -> 
( R1 `  x
)  e.  T )  <-> 
( ( T  e. 
Tarski  /\  T  =/=  (/) )  -> 
y  e.  T ) ) )
2623, 25syl5ibcom 212 . . . . 5  |-  ( x  e.  om  ->  (
( R1 `  x
)  =  y  -> 
( ( T  e. 
Tarski  /\  T  =/=  (/) )  -> 
y  e.  T ) ) )
2726rexlimiv 2824 . . . 4  |-  ( E. x  e.  om  ( R1 `  x )  =  y  ->  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  y  e.  T ) )
285, 27syl 16 . . 3  |-  ( y  e.  ( R1 " om )  ->  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  y  e.  T ) )
2928com12 29 . 2  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  (
y  e.  ( R1
" om )  -> 
y  e.  T ) )
3029ssrdv 3354 1  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  ( R1 " om )  C_  T )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2599   E.wrex 2706    C_ wss 3320   (/)c0 3628   ~Pcpw 3799   Oncon0 4581   suc csuc 4583   omcom 4845   "cima 4881   Fun wfun 5448    Fn wfn 5449   ` cfv 5454   R1cr1 7688   Tarskictsk 8623
This theorem is referenced by:  tskr1om2  8643  tskinf  8644
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-recs 6633  df-rdg 6668  df-r1 7690  df-tsk 8624
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