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Theorem tskr1om2 8931
Description: A nonempty Tarski class contains the whole finite cumulative hierarchy. (This proof does not use ax-inf 7840.) (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 4092 . . 3  |-  ( y  e.  U. ( R1
" om )  <->  E. x  e.  ( R1 " om ) y  e.  x
)
2 r1fnon 7970 . . . . . . . . 9  |-  R1  Fn  On
3 fnfun 5505 . . . . . . . . 9  |-  ( R1  Fn  On  ->  Fun  R1 )
42, 3ax-mp 5 . . . . . . . 8  |-  Fun  R1
5 fvelima 5740 . . . . . . . 8  |-  ( ( Fun  R1  /\  x  e.  ( R1 " om ) )  ->  E. y  e.  om  ( R1 `  y )  =  x )
64, 5mpan 665 . . . . . . 7  |-  ( x  e.  ( R1 " om )  ->  E. y  e.  om  ( R1 `  y )  =  x )
7 r1tr 7979 . . . . . . . . 9  |-  Tr  ( R1 `  y )
8 treq 4388 . . . . . . . . 9  |-  ( ( R1 `  y )  =  x  ->  ( Tr  ( R1 `  y
)  <->  Tr  x )
)
97, 8mpbii 211 . . . . . . . 8  |-  ( ( R1 `  y )  =  x  ->  Tr  x )
109rexlimivw 2835 . . . . . . 7  |-  ( E. y  e.  om  ( R1 `  y )  =  x  ->  Tr  x
)
11 trss 4391 . . . . . . 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 463 . . . . 5  |-  ( ( ( T  e.  Tarski  /\  T  =/=  (/) )  /\  x  e.  ( R1 " om ) )  -> 
( y  e.  x  ->  y  C_  x )
)
14 tskr1om 8930 . . . . . . . 8  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  ( R1 " om )  C_  T )
1514sseld 3352 . . . . . . 7  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  (
x  e.  ( R1
" om )  ->  x  e.  T )
)
16 tskss 8921 . . . . . . . . 9  |-  ( ( T  e.  Tarski  /\  x  e.  T  /\  y  C_  x )  ->  y  e.  T )
17163exp 1181 . . . . . . . 8  |-  ( T  e.  Tarski  ->  ( x  e.  T  ->  ( y  C_  x  ->  y  e.  T ) ) )
1817adantr 462 . . . . . . 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 2839 . . 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 3359 1  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  U. ( R1 " om )  C_  T )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1364    e. wcel 1761    =/= wne 2604   E.wrex 2714    C_ wss 3325   (/)c0 3634   U.cuni 4088   Tr wtr 4382   Oncon0 4715   "cima 4839   Fun wfun 5409    Fn wfn 5410   ` cfv 5415   omcom 6475   R1cr1 7965   Tarskictsk 8911
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-om 6476  df-recs 6828  df-rdg 6862  df-r1 7967  df-tsk 8912
This theorem is referenced by: (None)
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