MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tskr1om Unicode version

Theorem tskr1om 8389
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 7339.) (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 7439 . . . . . 6  |-  R1  Fn  On
2 fnfun 5341 . . . . . 6  |-  ( R1  Fn  On  ->  Fun  R1 )
31, 2ax-mp 8 . . . . 5  |-  Fun  R1
4 fvelima 5574 . . . . 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 5525 . . . . . . . 8  |-  ( x  =  (/)  ->  ( R1
`  x )  =  ( R1 `  (/) ) )
76eleq1d 2349 . . . . . . 7  |-  ( x  =  (/)  ->  ( ( R1 `  x )  e.  T  <->  ( R1 `  (/) )  e.  T
) )
8 fveq2 5525 . . . . . . . 8  |-  ( x  =  y  ->  ( R1 `  x )  =  ( R1 `  y
) )
98eleq1d 2349 . . . . . . 7  |-  ( x  =  y  ->  (
( R1 `  x
)  e.  T  <->  ( R1 `  y )  e.  T
) )
10 fveq2 5525 . . . . . . . 8  |-  ( x  =  suc  y  -> 
( R1 `  x
)  =  ( R1
`  suc  y )
)
1110eleq1d 2349 . . . . . . 7  |-  ( x  =  suc  y  -> 
( ( R1 `  x )  e.  T  <->  ( R1 `  suc  y
)  e.  T ) )
12 r10 7440 . . . . . . . 8  |-  ( R1
`  (/) )  =  (/)
13 tsk0 8385 . . . . . . . 8  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  (/)  e.  T
)
1412, 13syl5eqel 2367 . . . . . . 7  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  ( R1 `  (/) )  e.  T
)
15 tskpw 8375 . . . . . . . . . 10  |-  ( ( T  e.  Tarski  /\  ( R1 `  y )  e.  T )  ->  ~P ( R1 `  y )  e.  T )
16 nnon 4662 . . . . . . . . . . . 12  |-  ( y  e.  om  ->  y  e.  On )
17 r1suc 7442 . . . . . . . . . . . 12  |-  ( y  e.  On  ->  ( R1 `  suc  y )  =  ~P ( R1
`  y ) )
1816, 17syl 15 . . . . . . . . . . 11  |-  ( y  e.  om  ->  ( R1 `  suc  y )  =  ~P ( R1
`  y ) )
1918eleq1d 2349 . . . . . . . . . 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 4684 . . . . . 6  |-  ( x  e.  om  ->  (
( T  e.  Tarski  /\  T  =/=  (/) )  -> 
( R1 `  x
)  e.  T ) )
24 eleq1 2343 . . . . . . 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 2661 . . . 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 3185 1  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  ( R1 " om )  C_  T )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544    C_ wss 3152   (/)c0 3455   ~Pcpw 3625   Oncon0 4392   suc csuc 4394   omcom 4656   "cima 4692   Fun wfun 5249    Fn wfn 5250   ` cfv 5255   R1cr1 7434   Tarskictsk 8370
This theorem is referenced by:  tskr1om2  8390  tskinf  8391
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-recs 6388  df-rdg 6423  df-r1 7436  df-tsk 8371
  Copyright terms: Public domain W3C validator