MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tskr1om2 Unicode version

Theorem tskr1om2 8390
Description: A nonempty Tarski's class contains the whole finite cumulative hierarchy. (This proof does not use ax-inf 7339.) (Contributed by NM, 22-Feb-2011.)
Assertion
Ref Expression
tskr1om2  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  U. ( R1 " om )  C_  T )

Proof of Theorem tskr1om2
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eluni2 3831 . . 3  |-  ( y  e.  U. ( R1
" om )  <->  E. x  e.  ( R1 " om ) y  e.  x
)
2 r1fnon 7439 . . . . . . . . 9  |-  R1  Fn  On
3 fnfun 5341 . . . . . . . . 9  |-  ( R1  Fn  On  ->  Fun  R1 )
42, 3ax-mp 8 . . . . . . . 8  |-  Fun  R1
5 fvelima 5574 . . . . . . . 8  |-  ( ( Fun  R1  /\  x  e.  ( R1 " om ) )  ->  E. y  e.  om  ( R1 `  y )  =  x )
64, 5mpan 651 . . . . . . 7  |-  ( x  e.  ( R1 " om )  ->  E. y  e.  om  ( R1 `  y )  =  x )
7 r1tr 7448 . . . . . . . . 9  |-  Tr  ( R1 `  y )
8 treq 4119 . . . . . . . . 9  |-  ( ( R1 `  y )  =  x  ->  ( Tr  ( R1 `  y
)  <->  Tr  x )
)
97, 8mpbii 202 . . . . . . . 8  |-  ( ( R1 `  y )  =  x  ->  Tr  x )
109rexlimivw 2663 . . . . . . 7  |-  ( E. y  e.  om  ( R1 `  y )  =  x  ->  Tr  x
)
11 trss 4122 . . . . . . 7  |-  ( Tr  x  ->  ( y  e.  x  ->  y  C_  x ) )
126, 10, 113syl 18 . . . . . 6  |-  ( x  e.  ( R1 " om )  ->  ( y  e.  x  ->  y  C_  x ) )
1312adantl 452 . . . . 5  |-  ( ( ( T  e.  Tarski  /\  T  =/=  (/) )  /\  x  e.  ( R1 " om ) )  -> 
( y  e.  x  ->  y  C_  x )
)
14 tskr1om 8389 . . . . . . . 8  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  ( R1 " om )  C_  T )
1514sseld 3179 . . . . . . 7  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  (
x  e.  ( R1
" om )  ->  x  e.  T )
)
16 tskss 8380 . . . . . . . . 9  |-  ( ( T  e.  Tarski  /\  x  e.  T  /\  y  C_  x )  ->  y  e.  T )
17163exp 1150 . . . . . . . 8  |-  ( T  e.  Tarski  ->  ( x  e.  T  ->  ( y  C_  x  ->  y  e.  T ) ) )
1817adantr 451 . . . . . . 7  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  (
x  e.  T  -> 
( y  C_  x  ->  y  e.  T ) ) )
1915, 18syld 40 . . . . . 6  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  (
x  e.  ( R1
" om )  -> 
( y  C_  x  ->  y  e.  T ) ) )
2019imp 418 . . . . 5  |-  ( ( ( T  e.  Tarski  /\  T  =/=  (/) )  /\  x  e.  ( R1 " om ) )  -> 
( y  C_  x  ->  y  e.  T ) )
2113, 20syld 40 . . . 4  |-  ( ( ( T  e.  Tarski  /\  T  =/=  (/) )  /\  x  e.  ( R1 " om ) )  -> 
( y  e.  x  ->  y  e.  T ) )
2221rexlimdva 2667 . . 3  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  ( E. x  e.  ( R1 " om ) y  e.  x  ->  y  e.  T ) )
231, 22syl5bi 208 . 2  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  (
y  e.  U. ( R1 " om )  -> 
y  e.  T ) )
2423ssrdv 3185 1  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  U. ( 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   U.cuni 3827   Tr wtr 4113   Oncon0 4392   omcom 4656   "cima 4692   Fun wfun 5249    Fn wfn 5250   ` cfv 5255   R1cr1 7434   Tarskictsk 8370
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