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Theorem tskr1om2 9144
Description: A nonempty Tarski class contains the whole finite cumulative hierarchy. (This proof does not use ax-inf 8053.) (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 4234 . . 3  |-  ( y  e.  U. ( R1
" om )  <->  E. x  e.  ( R1 " om ) y  e.  x
)
2 r1fnon 8183 . . . . . . . . 9  |-  R1  Fn  On
3 fnfun 5664 . . . . . . . . 9  |-  ( R1  Fn  On  ->  Fun  R1 )
42, 3ax-mp 5 . . . . . . . 8  |-  Fun  R1
5 fvelima 5906 . . . . . . . 8  |-  ( ( Fun  R1  /\  x  e.  ( R1 " om ) )  ->  E. y  e.  om  ( R1 `  y )  =  x )
64, 5mpan 670 . . . . . . 7  |-  ( x  e.  ( R1 " om )  ->  E. y  e.  om  ( R1 `  y )  =  x )
7 r1tr 8192 . . . . . . . . 9  |-  Tr  ( R1 `  y )
8 treq 4532 . . . . . . . . 9  |-  ( ( R1 `  y )  =  x  ->  ( Tr  ( R1 `  y
)  <->  Tr  x )
)
97, 8mpbii 211 . . . . . . . 8  |-  ( ( R1 `  y )  =  x  ->  Tr  x )
109rexlimivw 2930 . . . . . . 7  |-  ( E. y  e.  om  ( R1 `  y )  =  x  ->  Tr  x
)
11 trss 4535 . . . . . . 7  |-  ( Tr  x  ->  ( y  e.  x  ->  y  C_  x ) )
126, 10, 113syl 20 . . . . . 6  |-  ( x  e.  ( R1 " om )  ->  ( y  e.  x  ->  y  C_  x ) )
1312adantl 466 . . . . 5  |-  ( ( ( T  e.  Tarski  /\  T  =/=  (/) )  /\  x  e.  ( R1 " om ) )  -> 
( y  e.  x  ->  y  C_  x )
)
14 tskr1om 9143 . . . . . . . 8  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  ( R1 " om )  C_  T )
1514sseld 3485 . . . . . . 7  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  (
x  e.  ( R1
" om )  ->  x  e.  T )
)
16 tskss 9134 . . . . . . . . 9  |-  ( ( T  e.  Tarski  /\  x  e.  T  /\  y  C_  x )  ->  y  e.  T )
17163exp 1194 . . . . . . . 8  |-  ( T  e.  Tarski  ->  ( x  e.  T  ->  ( y  C_  x  ->  y  e.  T ) ) )
1817adantr 465 . . . . . . 7  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  (
x  e.  T  -> 
( y  C_  x  ->  y  e.  T ) ) )
1915, 18syld 44 . . . . . 6  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  (
x  e.  ( R1
" om )  -> 
( y  C_  x  ->  y  e.  T ) ) )
2019imp 429 . . . . 5  |-  ( ( ( T  e.  Tarski  /\  T  =/=  (/) )  /\  x  e.  ( R1 " om ) )  -> 
( y  C_  x  ->  y  e.  T ) )
2113, 20syld 44 . . . 4  |-  ( ( ( T  e.  Tarski  /\  T  =/=  (/) )  /\  x  e.  ( R1 " om ) )  -> 
( y  e.  x  ->  y  e.  T ) )
2221rexlimdva 2933 . . 3  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  ( E. x  e.  ( R1 " om ) y  e.  x  ->  y  e.  T ) )
231, 22syl5bi 217 . 2  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  (
y  e.  U. ( R1 " om )  -> 
y  e.  T ) )
2423ssrdv 3492 1  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  U. ( R1 " om )  C_  T )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1381    e. wcel 1802    =/= wne 2636   E.wrex 2792    C_ wss 3458   (/)c0 3767   U.cuni 4230   Tr wtr 4526   Oncon0 4864   "cima 4988   Fun wfun 5568    Fn wfn 5569   ` cfv 5574   omcom 6681   R1cr1 8178   Tarskictsk 9124
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-reu 2798  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-om 6682  df-recs 7040  df-rdg 7074  df-r1 8180  df-tsk 9125
This theorem is referenced by: (None)
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