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Theorem tskord 9159
Description: A Tarski class contains all ordinals smaller than it. (Contributed by Mario Carneiro, 8-Jun-2013.)
Assertion
Ref Expression
tskord  |-  ( ( T  e.  Tarski  /\  A  e.  On  /\  A  ~<  T )  ->  A  e.  T )

Proof of Theorem tskord
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq1 4450 . . . . . 6  |-  ( x  =  y  ->  (
x  ~<  T  <->  y  ~<  T ) )
21anbi2d 703 . . . . 5  |-  ( x  =  y  ->  (
( T  e.  Tarski  /\  x  ~<  T )  <->  ( T  e.  Tarski  /\  y  ~<  T ) ) )
3 eleq1 2539 . . . . 5  |-  ( x  =  y  ->  (
x  e.  T  <->  y  e.  T ) )
42, 3imbi12d 320 . . . 4  |-  ( x  =  y  ->  (
( ( T  e. 
Tarski  /\  x  ~<  T )  ->  x  e.  T
)  <->  ( ( T  e.  Tarski  /\  y  ~<  T )  ->  y  e.  T ) ) )
5 breq1 4450 . . . . . 6  |-  ( x  =  A  ->  (
x  ~<  T  <->  A  ~<  T ) )
65anbi2d 703 . . . . 5  |-  ( x  =  A  ->  (
( T  e.  Tarski  /\  x  ~<  T )  <->  ( T  e.  Tarski  /\  A  ~<  T ) ) )
7 eleq1 2539 . . . . 5  |-  ( x  =  A  ->  (
x  e.  T  <->  A  e.  T ) )
86, 7imbi12d 320 . . . 4  |-  ( x  =  A  ->  (
( ( T  e. 
Tarski  /\  x  ~<  T )  ->  x  e.  T
)  <->  ( ( T  e.  Tarski  /\  A  ~<  T )  ->  A  e.  T ) ) )
9 simplrl 759 . . . . . . . . 9  |-  ( ( ( x  e.  On  /\  ( T  e.  Tarski  /\  x  ~<  T )
)  /\  y  e.  x )  ->  T  e.  Tarski )
10 onelss 4920 . . . . . . . . . . . . 13  |-  ( x  e.  On  ->  (
y  e.  x  -> 
y  C_  x )
)
11 ssdomg 7562 . . . . . . . . . . . . 13  |-  ( x  e.  On  ->  (
y  C_  x  ->  y  ~<_  x ) )
1210, 11syld 44 . . . . . . . . . . . 12  |-  ( x  e.  On  ->  (
y  e.  x  -> 
y  ~<_  x ) )
1312imp 429 . . . . . . . . . . 11  |-  ( ( x  e.  On  /\  y  e.  x )  ->  y  ~<_  x )
1413adantlr 714 . . . . . . . . . 10  |-  ( ( ( x  e.  On  /\  ( T  e.  Tarski  /\  x  ~<  T )
)  /\  y  e.  x )  ->  y  ~<_  x )
15 simplrr 760 . . . . . . . . . 10  |-  ( ( ( x  e.  On  /\  ( T  e.  Tarski  /\  x  ~<  T )
)  /\  y  e.  x )  ->  x  ~<  T )
16 domsdomtr 7653 . . . . . . . . . 10  |-  ( ( y  ~<_  x  /\  x  ~<  T )  ->  y  ~<  T )
1714, 15, 16syl2anc 661 . . . . . . . . 9  |-  ( ( ( x  e.  On  /\  ( T  e.  Tarski  /\  x  ~<  T )
)  /\  y  e.  x )  ->  y  ~<  T )
18 pm2.27 39 . . . . . . . . 9  |-  ( ( T  e.  Tarski  /\  y  ~<  T )  ->  (
( ( T  e. 
Tarski  /\  y  ~<  T )  ->  y  e.  T
)  ->  y  e.  T ) )
199, 17, 18syl2anc 661 . . . . . . . 8  |-  ( ( ( x  e.  On  /\  ( T  e.  Tarski  /\  x  ~<  T )
)  /\  y  e.  x )  ->  (
( ( T  e. 
Tarski  /\  y  ~<  T )  ->  y  e.  T
)  ->  y  e.  T ) )
2019ralimdva 2872 . . . . . . 7  |-  ( ( x  e.  On  /\  ( T  e.  Tarski  /\  x  ~<  T ) )  -> 
( A. y  e.  x  ( ( T  e.  Tarski  /\  y  ~<  T )  ->  y  e.  T )  ->  A. y  e.  x  y  e.  T ) )
21 dfss3 3494 . . . . . . . . . . 11  |-  ( x 
C_  T  <->  A. y  e.  x  y  e.  T )
22 tskssel 9136 . . . . . . . . . . . 12  |-  ( ( T  e.  Tarski  /\  x  C_  T  /\  x  ~<  T )  ->  x  e.  T )
23223exp 1195 . . . . . . . . . . 11  |-  ( T  e.  Tarski  ->  ( x  C_  T  ->  ( x  ~<  T  ->  x  e.  T
) ) )
2421, 23syl5bir 218 . . . . . . . . . 10  |-  ( T  e.  Tarski  ->  ( A. y  e.  x  y  e.  T  ->  ( x  ~<  T  ->  x  e.  T
) ) )
2524com23 78 . . . . . . . . 9  |-  ( T  e.  Tarski  ->  ( x  ~<  T  ->  ( A. y  e.  x  y  e.  T  ->  x  e.  T
) ) )
2625imp 429 . . . . . . . 8  |-  ( ( T  e.  Tarski  /\  x  ~<  T )  ->  ( A. y  e.  x  y  e.  T  ->  x  e.  T ) )
2726adantl 466 . . . . . . 7  |-  ( ( x  e.  On  /\  ( T  e.  Tarski  /\  x  ~<  T ) )  -> 
( A. y  e.  x  y  e.  T  ->  x  e.  T ) )
2820, 27syld 44 . . . . . 6  |-  ( ( x  e.  On  /\  ( T  e.  Tarski  /\  x  ~<  T ) )  -> 
( A. y  e.  x  ( ( T  e.  Tarski  /\  y  ~<  T )  ->  y  e.  T )  ->  x  e.  T ) )
2928ex 434 . . . . 5  |-  ( x  e.  On  ->  (
( T  e.  Tarski  /\  x  ~<  T )  ->  ( A. y  e.  x  ( ( T  e.  Tarski  /\  y  ~<  T )  ->  y  e.  T )  ->  x  e.  T ) ) )
3029com23 78 . . . 4  |-  ( x  e.  On  ->  ( A. y  e.  x  ( ( T  e. 
Tarski  /\  y  ~<  T )  ->  y  e.  T
)  ->  ( ( T  e.  Tarski  /\  x  ~<  T )  ->  x  e.  T ) ) )
314, 8, 30tfis3 6677 . . 3  |-  ( A  e.  On  ->  (
( T  e.  Tarski  /\  A  ~<  T )  ->  A  e.  T ) )
32313impib 1194 . 2  |-  ( ( A  e.  On  /\  T  e.  Tarski  /\  A  ~<  T )  ->  A  e.  T )
33323com12 1200 1  |-  ( ( T  e.  Tarski  /\  A  e.  On  /\  A  ~<  T )  ->  A  e.  T )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2814    C_ wss 3476   class class class wbr 4447   Oncon0 4878    ~<_ cdom 7515    ~< csdm 7516   Tarskictsk 9127
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-er 7312  df-en 7518  df-dom 7519  df-sdom 7520  df-tsk 9128
This theorem is referenced by:  tskcard  9160
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