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Theorem tskord 9194
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 4420 . . . . . 6  |-  ( x  =  y  ->  (
x  ~<  T  <->  y  ~<  T ) )
21anbi2d 708 . . . . 5  |-  ( x  =  y  ->  (
( T  e.  Tarski  /\  x  ~<  T )  <->  ( T  e.  Tarski  /\  y  ~<  T ) ) )
3 eleq1 2492 . . . . 5  |-  ( x  =  y  ->  (
x  e.  T  <->  y  e.  T ) )
42, 3imbi12d 321 . . . 4  |-  ( x  =  y  ->  (
( ( T  e. 
Tarski  /\  x  ~<  T )  ->  x  e.  T
)  <->  ( ( T  e.  Tarski  /\  y  ~<  T )  ->  y  e.  T ) ) )
5 breq1 4420 . . . . . 6  |-  ( x  =  A  ->  (
x  ~<  T  <->  A  ~<  T ) )
65anbi2d 708 . . . . 5  |-  ( x  =  A  ->  (
( T  e.  Tarski  /\  x  ~<  T )  <->  ( T  e.  Tarski  /\  A  ~<  T ) ) )
7 eleq1 2492 . . . . 5  |-  ( x  =  A  ->  (
x  e.  T  <->  A  e.  T ) )
86, 7imbi12d 321 . . . 4  |-  ( x  =  A  ->  (
( ( T  e. 
Tarski  /\  x  ~<  T )  ->  x  e.  T
)  <->  ( ( T  e.  Tarski  /\  A  ~<  T )  ->  A  e.  T ) ) )
9 simplrl 768 . . . . . . . . 9  |-  ( ( ( x  e.  On  /\  ( T  e.  Tarski  /\  x  ~<  T )
)  /\  y  e.  x )  ->  T  e.  Tarski )
10 onelss 5475 . . . . . . . . . . . . 13  |-  ( x  e.  On  ->  (
y  e.  x  -> 
y  C_  x )
)
11 ssdomg 7613 . . . . . . . . . . . . 13  |-  ( x  e.  On  ->  (
y  C_  x  ->  y  ~<_  x ) )
1210, 11syld 45 . . . . . . . . . . . 12  |-  ( x  e.  On  ->  (
y  e.  x  -> 
y  ~<_  x ) )
1312imp 430 . . . . . . . . . . 11  |-  ( ( x  e.  On  /\  y  e.  x )  ->  y  ~<_  x )
1413adantlr 719 . . . . . . . . . 10  |-  ( ( ( x  e.  On  /\  ( T  e.  Tarski  /\  x  ~<  T )
)  /\  y  e.  x )  ->  y  ~<_  x )
15 simplrr 769 . . . . . . . . . 10  |-  ( ( ( x  e.  On  /\  ( T  e.  Tarski  /\  x  ~<  T )
)  /\  y  e.  x )  ->  x  ~<  T )
16 domsdomtr 7704 . . . . . . . . . 10  |-  ( ( y  ~<_  x  /\  x  ~<  T )  ->  y  ~<  T )
1714, 15, 16syl2anc 665 . . . . . . . . 9  |-  ( ( ( x  e.  On  /\  ( T  e.  Tarski  /\  x  ~<  T )
)  /\  y  e.  x )  ->  y  ~<  T )
18 pm2.27 40 . . . . . . . . 9  |-  ( ( T  e.  Tarski  /\  y  ~<  T )  ->  (
( ( T  e. 
Tarski  /\  y  ~<  T )  ->  y  e.  T
)  ->  y  e.  T ) )
199, 17, 18syl2anc 665 . . . . . . . 8  |-  ( ( ( x  e.  On  /\  ( T  e.  Tarski  /\  x  ~<  T )
)  /\  y  e.  x )  ->  (
( ( T  e. 
Tarski  /\  y  ~<  T )  ->  y  e.  T
)  ->  y  e.  T ) )
2019ralimdva 2831 . . . . . . 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 3451 . . . . . . . . . . 11  |-  ( x 
C_  T  <->  A. y  e.  x  y  e.  T )
22 tskssel 9171 . . . . . . . . . . . 12  |-  ( ( T  e.  Tarski  /\  x  C_  T  /\  x  ~<  T )  ->  x  e.  T )
23223exp 1204 . . . . . . . . . . 11  |-  ( T  e.  Tarski  ->  ( x  C_  T  ->  ( x  ~<  T  ->  x  e.  T
) ) )
2421, 23syl5bir 221 . . . . . . . . . 10  |-  ( T  e.  Tarski  ->  ( A. y  e.  x  y  e.  T  ->  ( x  ~<  T  ->  x  e.  T
) ) )
2524com23 81 . . . . . . . . 9  |-  ( T  e.  Tarski  ->  ( x  ~<  T  ->  ( A. y  e.  x  y  e.  T  ->  x  e.  T
) ) )
2625imp 430 . . . . . . . 8  |-  ( ( T  e.  Tarski  /\  x  ~<  T )  ->  ( A. y  e.  x  y  e.  T  ->  x  e.  T ) )
2726adantl 467 . . . . . . 7  |-  ( ( x  e.  On  /\  ( T  e.  Tarski  /\  x  ~<  T ) )  -> 
( A. y  e.  x  y  e.  T  ->  x  e.  T ) )
2820, 27syld 45 . . . . . 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 435 . . . . 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 81 . . . 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 6689 . . 3  |-  ( A  e.  On  ->  (
( T  e.  Tarski  /\  A  ~<  T )  ->  A  e.  T ) )
32313impib 1203 . 2  |-  ( ( A  e.  On  /\  T  e.  Tarski  /\  A  ~<  T )  ->  A  e.  T )
33323com12 1209 1  |-  ( ( T  e.  Tarski  /\  A  e.  On  /\  A  ~<  T )  ->  A  e.  T )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1867   A.wral 2773    C_ wss 3433   class class class wbr 4417   Oncon0 5433    ~<_ cdom 7566    ~< csdm 7567   Tarskictsk 9162
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-sbc 3297  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-br 4418  df-opab 4476  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-we 4806  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-ord 5436  df-on 5437  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-er 7362  df-en 7569  df-dom 7570  df-sdom 7571  df-tsk 9163
This theorem is referenced by:  tskcard  9195
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