MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tskord Structured version   Unicode version

Theorem tskord 8647
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 4207 . . . . . 6  |-  ( x  =  y  ->  (
x  ~<  T  <->  y  ~<  T ) )
21anbi2d 685 . . . . 5  |-  ( x  =  y  ->  (
( T  e.  Tarski  /\  x  ~<  T )  <->  ( T  e.  Tarski  /\  y  ~<  T ) ) )
3 eleq1 2495 . . . . 5  |-  ( x  =  y  ->  (
x  e.  T  <->  y  e.  T ) )
42, 3imbi12d 312 . . . 4  |-  ( x  =  y  ->  (
( ( T  e. 
Tarski  /\  x  ~<  T )  ->  x  e.  T
)  <->  ( ( T  e.  Tarski  /\  y  ~<  T )  ->  y  e.  T ) ) )
5 breq1 4207 . . . . . 6  |-  ( x  =  A  ->  (
x  ~<  T  <->  A  ~<  T ) )
65anbi2d 685 . . . . 5  |-  ( x  =  A  ->  (
( T  e.  Tarski  /\  x  ~<  T )  <->  ( T  e.  Tarski  /\  A  ~<  T ) ) )
7 eleq1 2495 . . . . 5  |-  ( x  =  A  ->  (
x  e.  T  <->  A  e.  T ) )
86, 7imbi12d 312 . . . 4  |-  ( x  =  A  ->  (
( ( T  e. 
Tarski  /\  x  ~<  T )  ->  x  e.  T
)  <->  ( ( T  e.  Tarski  /\  A  ~<  T )  ->  A  e.  T ) ) )
9 simplrl 737 . . . . . . . . 9  |-  ( ( ( x  e.  On  /\  ( T  e.  Tarski  /\  x  ~<  T )
)  /\  y  e.  x )  ->  T  e.  Tarski )
10 onelss 4615 . . . . . . . . . . . . 13  |-  ( x  e.  On  ->  (
y  e.  x  -> 
y  C_  x )
)
11 ssdomg 7145 . . . . . . . . . . . . 13  |-  ( x  e.  On  ->  (
y  C_  x  ->  y  ~<_  x ) )
1210, 11syld 42 . . . . . . . . . . . 12  |-  ( x  e.  On  ->  (
y  e.  x  -> 
y  ~<_  x ) )
1312imp 419 . . . . . . . . . . 11  |-  ( ( x  e.  On  /\  y  e.  x )  ->  y  ~<_  x )
1413adantlr 696 . . . . . . . . . 10  |-  ( ( ( x  e.  On  /\  ( T  e.  Tarski  /\  x  ~<  T )
)  /\  y  e.  x )  ->  y  ~<_  x )
15 simplrr 738 . . . . . . . . . 10  |-  ( ( ( x  e.  On  /\  ( T  e.  Tarski  /\  x  ~<  T )
)  /\  y  e.  x )  ->  x  ~<  T )
16 domsdomtr 7234 . . . . . . . . . 10  |-  ( ( y  ~<_  x  /\  x  ~<  T )  ->  y  ~<  T )
1714, 15, 16syl2anc 643 . . . . . . . . 9  |-  ( ( ( x  e.  On  /\  ( T  e.  Tarski  /\  x  ~<  T )
)  /\  y  e.  x )  ->  y  ~<  T )
18 pm2.27 37 . . . . . . . . 9  |-  ( ( T  e.  Tarski  /\  y  ~<  T )  ->  (
( ( T  e. 
Tarski  /\  y  ~<  T )  ->  y  e.  T
)  ->  y  e.  T ) )
199, 17, 18syl2anc 643 . . . . . . . 8  |-  ( ( ( x  e.  On  /\  ( T  e.  Tarski  /\  x  ~<  T )
)  /\  y  e.  x )  ->  (
( ( T  e. 
Tarski  /\  y  ~<  T )  ->  y  e.  T
)  ->  y  e.  T ) )
2019ralimdva 2776 . . . . . . 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 3330 . . . . . . . . . . 11  |-  ( x 
C_  T  <->  A. y  e.  x  y  e.  T )
22 tskssel 8624 . . . . . . . . . . . 12  |-  ( ( T  e.  Tarski  /\  x  C_  T  /\  x  ~<  T )  ->  x  e.  T )
23223exp 1152 . . . . . . . . . . 11  |-  ( T  e.  Tarski  ->  ( x  C_  T  ->  ( x  ~<  T  ->  x  e.  T
) ) )
2421, 23syl5bir 210 . . . . . . . . . 10  |-  ( T  e.  Tarski  ->  ( A. y  e.  x  y  e.  T  ->  ( x  ~<  T  ->  x  e.  T
) ) )
2524com23 74 . . . . . . . . 9  |-  ( T  e.  Tarski  ->  ( x  ~<  T  ->  ( A. y  e.  x  y  e.  T  ->  x  e.  T
) ) )
2625imp 419 . . . . . . . 8  |-  ( ( T  e.  Tarski  /\  x  ~<  T )  ->  ( A. y  e.  x  y  e.  T  ->  x  e.  T ) )
2726adantl 453 . . . . . . 7  |-  ( ( x  e.  On  /\  ( T  e.  Tarski  /\  x  ~<  T ) )  -> 
( A. y  e.  x  y  e.  T  ->  x  e.  T ) )
2820, 27syld 42 . . . . . 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 424 . . . . 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 74 . . . 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 4829 . . 3  |-  ( A  e.  On  ->  (
( T  e.  Tarski  /\  A  ~<  T )  ->  A  e.  T ) )
32313impib 1151 . 2  |-  ( ( A  e.  On  /\  T  e.  Tarski  /\  A  ~<  T )  ->  A  e.  T )
33323com12 1157 1  |-  ( ( T  e.  Tarski  /\  A  e.  On  /\  A  ~<  T )  ->  A  e.  T )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2697    C_ wss 3312   class class class wbr 4204   Oncon0 4573    ~<_ cdom 7099    ~< csdm 7100   Tarskictsk 8615
This theorem is referenced by:  tskcard  8648
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-tsk 8616
  Copyright terms: Public domain W3C validator