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Theorem tskr1om2 8644
Description: A nonempty Tarski's class contains the whole finite cumulative hierarchy. (This proof does not use ax-inf 7594.) (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 4020 . . 3  |-  ( y  e.  U. ( R1
" om )  <->  E. x  e.  ( R1 " om ) y  e.  x
)
2 r1fnon 7694 . . . . . . . . 9  |-  R1  Fn  On
3 fnfun 5543 . . . . . . . . 9  |-  ( R1  Fn  On  ->  Fun  R1 )
42, 3ax-mp 8 . . . . . . . 8  |-  Fun  R1
5 fvelima 5779 . . . . . . . 8  |-  ( ( Fun  R1  /\  x  e.  ( R1 " om ) )  ->  E. y  e.  om  ( R1 `  y )  =  x )
64, 5mpan 653 . . . . . . 7  |-  ( x  e.  ( R1 " om )  ->  E. y  e.  om  ( R1 `  y )  =  x )
7 r1tr 7703 . . . . . . . . 9  |-  Tr  ( R1 `  y )
8 treq 4309 . . . . . . . . 9  |-  ( ( R1 `  y )  =  x  ->  ( Tr  ( R1 `  y
)  <->  Tr  x )
)
97, 8mpbii 204 . . . . . . . 8  |-  ( ( R1 `  y )  =  x  ->  Tr  x )
109rexlimivw 2827 . . . . . . 7  |-  ( E. y  e.  om  ( R1 `  y )  =  x  ->  Tr  x
)
11 trss 4312 . . . . . . 7  |-  ( Tr  x  ->  ( y  e.  x  ->  y  C_  x ) )
126, 10, 113syl 19 . . . . . 6  |-  ( x  e.  ( R1 " om )  ->  ( y  e.  x  ->  y  C_  x ) )
1312adantl 454 . . . . 5  |-  ( ( ( T  e.  Tarski  /\  T  =/=  (/) )  /\  x  e.  ( R1 " om ) )  -> 
( y  e.  x  ->  y  C_  x )
)
14 tskr1om 8643 . . . . . . . 8  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  ( R1 " om )  C_  T )
1514sseld 3348 . . . . . . 7  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  (
x  e.  ( R1
" om )  ->  x  e.  T )
)
16 tskss 8634 . . . . . . . . 9  |-  ( ( T  e.  Tarski  /\  x  e.  T  /\  y  C_  x )  ->  y  e.  T )
17163exp 1153 . . . . . . . 8  |-  ( T  e.  Tarski  ->  ( x  e.  T  ->  ( y  C_  x  ->  y  e.  T ) ) )
1817adantr 453 . . . . . . 7  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  (
x  e.  T  -> 
( y  C_  x  ->  y  e.  T ) ) )
1915, 18syld 43 . . . . . 6  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  (
x  e.  ( R1
" om )  -> 
( y  C_  x  ->  y  e.  T ) ) )
2019imp 420 . . . . 5  |-  ( ( ( T  e.  Tarski  /\  T  =/=  (/) )  /\  x  e.  ( R1 " om ) )  -> 
( y  C_  x  ->  y  e.  T ) )
2113, 20syld 43 . . . 4  |-  ( ( ( T  e.  Tarski  /\  T  =/=  (/) )  /\  x  e.  ( R1 " om ) )  -> 
( y  e.  x  ->  y  e.  T ) )
2221rexlimdva 2831 . . 3  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  ( E. x  e.  ( R1 " om ) y  e.  x  ->  y  e.  T ) )
231, 22syl5bi 210 . 2  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  (
y  e.  U. ( R1 " om )  -> 
y  e.  T ) )
2423ssrdv 3355 1  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  U. ( R1 " om )  C_  T )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2600   E.wrex 2707    C_ wss 3321   (/)c0 3629   U.cuni 4016   Tr wtr 4303   Oncon0 4582   omcom 4846   "cima 4882   Fun wfun 5449    Fn wfn 5450   ` cfv 5455   R1cr1 7689   Tarskictsk 8624
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-om 4847  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-recs 6634  df-rdg 6669  df-r1 7691  df-tsk 8625
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