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Theorem tskr1om2 9211
Description: A nonempty Tarski class contains the whole finite cumulative hierarchy. (This proof does not use ax-inf 8161.) (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 4194 . . 3  |-  ( y  e.  U. ( R1
" om )  <->  E. x  e.  ( R1 " om ) y  e.  x
)
2 r1fnon 8256 . . . . . . . . 9  |-  R1  Fn  On
3 fnfun 5683 . . . . . . . . 9  |-  ( R1  Fn  On  ->  Fun  R1 )
42, 3ax-mp 5 . . . . . . . 8  |-  Fun  R1
5 fvelima 5931 . . . . . . . 8  |-  ( ( Fun  R1  /\  x  e.  ( R1 " om ) )  ->  E. y  e.  om  ( R1 `  y )  =  x )
64, 5mpan 684 . . . . . . 7  |-  ( x  e.  ( R1 " om )  ->  E. y  e.  om  ( R1 `  y )  =  x )
7 r1tr 8265 . . . . . . . . 9  |-  Tr  ( R1 `  y )
8 treq 4496 . . . . . . . . 9  |-  ( ( R1 `  y )  =  x  ->  ( Tr  ( R1 `  y
)  <->  Tr  x )
)
97, 8mpbii 216 . . . . . . . 8  |-  ( ( R1 `  y )  =  x  ->  Tr  x )
109rexlimivw 2869 . . . . . . 7  |-  ( E. y  e.  om  ( R1 `  y )  =  x  ->  Tr  x
)
11 trss 4499 . . . . . . 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 473 . . . . 5  |-  ( ( ( T  e.  Tarski  /\  T  =/=  (/) )  /\  x  e.  ( R1 " om ) )  -> 
( y  e.  x  ->  y  C_  x )
)
14 tskr1om 9210 . . . . . . . 8  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  ( R1 " om )  C_  T )
1514sseld 3417 . . . . . . 7  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  (
x  e.  ( R1
" om )  ->  x  e.  T )
)
16 tskss 9201 . . . . . . . . 9  |-  ( ( T  e.  Tarski  /\  x  e.  T  /\  y  C_  x )  ->  y  e.  T )
17163exp 1230 . . . . . . . 8  |-  ( T  e.  Tarski  ->  ( x  e.  T  ->  ( y  C_  x  ->  y  e.  T ) ) )
1817adantr 472 . . . . . . 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 436 . . . . 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 2871 . . 3  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  ( E. x  e.  ( R1 " om ) y  e.  x  ->  y  e.  T ) )
231, 22syl5bi 225 . 2  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  (
y  e.  U. ( R1 " om )  -> 
y  e.  T ) )
2423ssrdv 3424 1  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  U. ( R1 " om )  C_  T )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376    = wceq 1452    e. wcel 1904    =/= wne 2641   E.wrex 2757    C_ wss 3390   (/)c0 3722   U.cuni 4190   Tr wtr 4490   "cima 4842   Oncon0 5430   Fun wfun 5583    Fn wfn 5584   ` cfv 5589   omcom 6711   R1cr1 8251   Tarskictsk 9191
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-om 6712  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-r1 8253  df-tsk 9192
This theorem is referenced by: (None)
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