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Theorem tskord 8418
Description: A Tarski's 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 4042 . . . . . 6  |-  ( x  =  y  ->  (
x  ~<  T  <->  y  ~<  T ) )
21anbi2d 684 . . . . 5  |-  ( x  =  y  ->  (
( T  e.  Tarski  /\  x  ~<  T )  <->  ( T  e.  Tarski  /\  y  ~<  T ) ) )
3 eleq1 2356 . . . . 5  |-  ( x  =  y  ->  (
x  e.  T  <->  y  e.  T ) )
42, 3imbi12d 311 . . . 4  |-  ( x  =  y  ->  (
( ( T  e. 
Tarski  /\  x  ~<  T )  ->  x  e.  T
)  <->  ( ( T  e.  Tarski  /\  y  ~<  T )  ->  y  e.  T ) ) )
5 breq1 4042 . . . . . 6  |-  ( x  =  A  ->  (
x  ~<  T  <->  A  ~<  T ) )
65anbi2d 684 . . . . 5  |-  ( x  =  A  ->  (
( T  e.  Tarski  /\  x  ~<  T )  <->  ( T  e.  Tarski  /\  A  ~<  T ) ) )
7 eleq1 2356 . . . . 5  |-  ( x  =  A  ->  (
x  e.  T  <->  A  e.  T ) )
86, 7imbi12d 311 . . . 4  |-  ( x  =  A  ->  (
( ( T  e. 
Tarski  /\  x  ~<  T )  ->  x  e.  T
)  <->  ( ( T  e.  Tarski  /\  A  ~<  T )  ->  A  e.  T ) ) )
9 simplrl 736 . . . . . . . . 9  |-  ( ( ( x  e.  On  /\  ( T  e.  Tarski  /\  x  ~<  T )
)  /\  y  e.  x )  ->  T  e.  Tarski )
10 onelss 4450 . . . . . . . . . . . . 13  |-  ( x  e.  On  ->  (
y  e.  x  -> 
y  C_  x )
)
11 ssdomg 6923 . . . . . . . . . . . . 13  |-  ( x  e.  On  ->  (
y  C_  x  ->  y  ~<_  x ) )
1210, 11syld 40 . . . . . . . . . . . 12  |-  ( x  e.  On  ->  (
y  e.  x  -> 
y  ~<_  x ) )
1312imp 418 . . . . . . . . . . 11  |-  ( ( x  e.  On  /\  y  e.  x )  ->  y  ~<_  x )
1413adantlr 695 . . . . . . . . . 10  |-  ( ( ( x  e.  On  /\  ( T  e.  Tarski  /\  x  ~<  T )
)  /\  y  e.  x )  ->  y  ~<_  x )
15 simplrr 737 . . . . . . . . . 10  |-  ( ( ( x  e.  On  /\  ( T  e.  Tarski  /\  x  ~<  T )
)  /\  y  e.  x )  ->  x  ~<  T )
16 domsdomtr 7012 . . . . . . . . . 10  |-  ( ( y  ~<_  x  /\  x  ~<  T )  ->  y  ~<  T )
1714, 15, 16syl2anc 642 . . . . . . . . 9  |-  ( ( ( x  e.  On  /\  ( T  e.  Tarski  /\  x  ~<  T )
)  /\  y  e.  x )  ->  y  ~<  T )
18 pm2.27 35 . . . . . . . . 9  |-  ( ( T  e.  Tarski  /\  y  ~<  T )  ->  (
( ( T  e. 
Tarski  /\  y  ~<  T )  ->  y  e.  T
)  ->  y  e.  T ) )
199, 17, 18syl2anc 642 . . . . . . . 8  |-  ( ( ( x  e.  On  /\  ( T  e.  Tarski  /\  x  ~<  T )
)  /\  y  e.  x )  ->  (
( ( T  e. 
Tarski  /\  y  ~<  T )  ->  y  e.  T
)  ->  y  e.  T ) )
2019ralimdva 2634 . . . . . . 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 3183 . . . . . . . . . . 11  |-  ( x 
C_  T  <->  A. y  e.  x  y  e.  T )
22 tskssel 8395 . . . . . . . . . . . 12  |-  ( ( T  e.  Tarski  /\  x  C_  T  /\  x  ~<  T )  ->  x  e.  T )
23223exp 1150 . . . . . . . . . . 11  |-  ( T  e.  Tarski  ->  ( x  C_  T  ->  ( x  ~<  T  ->  x  e.  T
) ) )
2421, 23syl5bir 209 . . . . . . . . . 10  |-  ( T  e.  Tarski  ->  ( A. y  e.  x  y  e.  T  ->  ( x  ~<  T  ->  x  e.  T
) ) )
2524com23 72 . . . . . . . . 9  |-  ( T  e.  Tarski  ->  ( x  ~<  T  ->  ( A. y  e.  x  y  e.  T  ->  x  e.  T
) ) )
2625imp 418 . . . . . . . 8  |-  ( ( T  e.  Tarski  /\  x  ~<  T )  ->  ( A. y  e.  x  y  e.  T  ->  x  e.  T ) )
2726adantl 452 . . . . . . 7  |-  ( ( x  e.  On  /\  ( T  e.  Tarski  /\  x  ~<  T ) )  -> 
( A. y  e.  x  y  e.  T  ->  x  e.  T ) )
2820, 27syld 40 . . . . . 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 423 . . . . 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 72 . . . 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 4664 . . 3  |-  ( A  e.  On  ->  (
( T  e.  Tarski  /\  A  ~<  T )  ->  A  e.  T ) )
32313impib 1149 . 2  |-  ( ( A  e.  On  /\  T  e.  Tarski  /\  A  ~<  T )  ->  A  e.  T )
33323com12 1155 1  |-  ( ( T  e.  Tarski  /\  A  e.  On  /\  A  ~<  T )  ->  A  e.  T )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556    C_ wss 3165   class class class wbr 4039   Oncon0 4408    ~<_ cdom 6877    ~< csdm 6878   Tarskictsk 8386
This theorem is referenced by:  tskcard  8419
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-tsk 8387
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