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Theorem tskr1om 9146
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 8056.) (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 8186 . . . . . 6  |-  R1  Fn  On
2 fnfun 5678 . . . . . 6  |-  ( R1  Fn  On  ->  Fun  R1 )
31, 2ax-mp 5 . . . . 5  |-  Fun  R1
4 fvelima 5920 . . . . 5  |-  ( ( Fun  R1  /\  y  e.  ( R1 " om ) )  ->  E. x  e.  om  ( R1 `  x )  =  y )
53, 4mpan 670 . . . 4  |-  ( y  e.  ( R1 " om )  ->  E. x  e.  om  ( R1 `  x )  =  y )
6 fveq2 5866 . . . . . . . 8  |-  ( x  =  (/)  ->  ( R1
`  x )  =  ( R1 `  (/) ) )
76eleq1d 2536 . . . . . . 7  |-  ( x  =  (/)  ->  ( ( R1 `  x )  e.  T  <->  ( R1 `  (/) )  e.  T
) )
8 fveq2 5866 . . . . . . . 8  |-  ( x  =  y  ->  ( R1 `  x )  =  ( R1 `  y
) )
98eleq1d 2536 . . . . . . 7  |-  ( x  =  y  ->  (
( R1 `  x
)  e.  T  <->  ( R1 `  y )  e.  T
) )
10 fveq2 5866 . . . . . . . 8  |-  ( x  =  suc  y  -> 
( R1 `  x
)  =  ( R1
`  suc  y )
)
1110eleq1d 2536 . . . . . . 7  |-  ( x  =  suc  y  -> 
( ( R1 `  x )  e.  T  <->  ( R1 `  suc  y
)  e.  T ) )
12 r10 8187 . . . . . . . 8  |-  ( R1
`  (/) )  =  (/)
13 tsk0 9142 . . . . . . . 8  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  (/)  e.  T
)
1412, 13syl5eqel 2559 . . . . . . 7  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  ( R1 `  (/) )  e.  T
)
15 tskpw 9132 . . . . . . . . . 10  |-  ( ( T  e.  Tarski  /\  ( R1 `  y )  e.  T )  ->  ~P ( R1 `  y )  e.  T )
16 nnon 6691 . . . . . . . . . . . 12  |-  ( y  e.  om  ->  y  e.  On )
17 r1suc 8189 . . . . . . . . . . . 12  |-  ( y  e.  On  ->  ( R1 `  suc  y )  =  ~P ( R1
`  y ) )
1816, 17syl 16 . . . . . . . . . . 11  |-  ( y  e.  om  ->  ( R1 `  suc  y )  =  ~P ( R1
`  y ) )
1918eleq1d 2536 . . . . . . . . . 10  |-  ( y  e.  om  ->  (
( R1 `  suc  y )  e.  T  <->  ~P ( R1 `  y
)  e.  T ) )
2015, 19syl5ibr 221 . . . . . . . . 9  |-  ( y  e.  om  ->  (
( T  e.  Tarski  /\  ( R1 `  y
)  e.  T )  ->  ( R1 `  suc  y )  e.  T
) )
2120expd 436 . . . . . . . 8  |-  ( y  e.  om  ->  ( T  e.  Tarski  ->  (
( R1 `  y
)  e.  T  -> 
( R1 `  suc  y )  e.  T
) ) )
2221adantrd 468 . . . . . . 7  |-  ( y  e.  om  ->  (
( T  e.  Tarski  /\  T  =/=  (/) )  -> 
( ( R1 `  y )  e.  T  ->  ( R1 `  suc  y )  e.  T
) ) )
237, 9, 11, 14, 22finds2 6713 . . . . . 6  |-  ( x  e.  om  ->  (
( T  e.  Tarski  /\  T  =/=  (/) )  -> 
( R1 `  x
)  e.  T ) )
24 eleq1 2539 . . . . . . 7  |-  ( ( R1 `  x )  =  y  ->  (
( R1 `  x
)  e.  T  <->  y  e.  T ) )
2524imbi2d 316 . . . . . 6  |-  ( ( R1 `  x )  =  y  ->  (
( ( T  e. 
Tarski  /\  T  =/=  (/) )  -> 
( R1 `  x
)  e.  T )  <-> 
( ( T  e. 
Tarski  /\  T  =/=  (/) )  -> 
y  e.  T ) ) )
2623, 25syl5ibcom 220 . . . . 5  |-  ( x  e.  om  ->  (
( R1 `  x
)  =  y  -> 
( ( T  e. 
Tarski  /\  T  =/=  (/) )  -> 
y  e.  T ) ) )
2726rexlimiv 2949 . . . 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 31 . 2  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  (
y  e.  ( R1
" om )  -> 
y  e.  T ) )
3029ssrdv 3510 1  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  ( R1 " om )  C_  T )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   E.wrex 2815    C_ wss 3476   (/)c0 3785   ~Pcpw 4010   Oncon0 4878   suc csuc 4880   "cima 5002   Fun wfun 5582    Fn wfn 5583   ` cfv 5588   omcom 6685   R1cr1 8181   Tarskictsk 9127
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-om 6686  df-recs 7043  df-rdg 7077  df-r1 8183  df-tsk 9128
This theorem is referenced by:  tskr1om2  9147  tskinf  9148
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