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Theorem cardmin2 7647
Description: The smallest ordinal that strictly dominates a set is a cardinal, if it exists. (Contributed by Mario Carneiro, 2-Feb-2013.)
Assertion
Ref Expression
cardmin2  |-  ( E. x  e.  On  A  ~<  x  <->  ( card `  |^| { x  e.  On  |  A  ~<  x } )  =  |^| { x  e.  On  |  A  ~<  x } )
Distinct variable group:    x, A

Proof of Theorem cardmin2
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 onintrab2 4609 . . . 4  |-  ( E. x  e.  On  A  ~<  x  <->  |^| { x  e.  On  |  A  ~<  x }  e.  On )
21biimpi 186 . . 3  |-  ( E. x  e.  On  A  ~<  x  ->  |^| { x  e.  On  |  A  ~<  x }  e.  On )
32adantr 451 . . . . . 6  |-  ( ( E. x  e.  On  A  ~<  x  /\  y  e.  |^| { x  e.  On  |  A  ~<  x } )  ->  |^| { x  e.  On  |  A  ~<  x }  e.  On )
4 eloni 4418 . . . . . . . 8  |-  ( |^| { x  e.  On  |  A  ~<  x }  e.  On  ->  Ord  |^| { x  e.  On  |  A  ~<  x } )
5 ordelss 4424 . . . . . . . 8  |-  ( ( Ord  |^| { x  e.  On  |  A  ~<  x }  /\  y  e. 
|^| { x  e.  On  |  A  ~<  x }
)  ->  y  C_  |^|
{ x  e.  On  |  A  ~<  x }
)
64, 5sylan 457 . . . . . . 7  |-  ( (
|^| { x  e.  On  |  A  ~<  x }  e.  On  /\  y  e. 
|^| { x  e.  On  |  A  ~<  x }
)  ->  y  C_  |^|
{ x  e.  On  |  A  ~<  x }
)
71, 6sylanb 458 . . . . . 6  |-  ( ( E. x  e.  On  A  ~<  x  /\  y  e.  |^| { x  e.  On  |  A  ~<  x } )  ->  y  C_ 
|^| { x  e.  On  |  A  ~<  x }
)
8 ssdomg 6923 . . . . . 6  |-  ( |^| { x  e.  On  |  A  ~<  x }  e.  On  ->  ( y  C_  |^|
{ x  e.  On  |  A  ~<  x }  ->  y  ~<_  |^| { x  e.  On  |  A  ~<  x } ) )
93, 7, 8sylc 56 . . . . 5  |-  ( ( E. x  e.  On  A  ~<  x  /\  y  e.  |^| { x  e.  On  |  A  ~<  x } )  ->  y  ~<_  |^|
{ x  e.  On  |  A  ~<  x }
)
10 onelon 4433 . . . . . . . 8  |-  ( (
|^| { x  e.  On  |  A  ~<  x }  e.  On  /\  y  e. 
|^| { x  e.  On  |  A  ~<  x }
)  ->  y  e.  On )
111, 10sylanb 458 . . . . . . 7  |-  ( ( E. x  e.  On  A  ~<  x  /\  y  e.  |^| { x  e.  On  |  A  ~<  x } )  ->  y  e.  On )
12 nfcv 2432 . . . . . . . . . . . . . 14  |-  F/_ x A
13 nfcv 2432 . . . . . . . . . . . . . 14  |-  F/_ x  ~<
14 nfrab1 2733 . . . . . . . . . . . . . . 15  |-  F/_ x { x  e.  On  |  A  ~<  x }
1514nfint 3888 . . . . . . . . . . . . . 14  |-  F/_ x |^| { x  e.  On  |  A  ~<  x }
1612, 13, 15nfbr 4083 . . . . . . . . . . . . 13  |-  F/ x  A  ~<  |^| { x  e.  On  |  A  ~<  x }
17 breq2 4043 . . . . . . . . . . . . 13  |-  ( x  =  |^| { x  e.  On  |  A  ~<  x }  ->  ( A  ~<  x  <->  A  ~<  |^| { x  e.  On  |  A  ~<  x } ) )
1816, 17onminsb 4606 . . . . . . . . . . . 12  |-  ( E. x  e.  On  A  ~<  x  ->  A  ~<  |^|
{ x  e.  On  |  A  ~<  x }
)
19 sdomentr 7011 . . . . . . . . . . . 12  |-  ( ( A  ~<  |^| { x  e.  On  |  A  ~<  x }  /\  |^| { x  e.  On  |  A  ~<  x }  ~~  y )  ->  A  ~<  y
)
2018, 19sylan 457 . . . . . . . . . . 11  |-  ( ( E. x  e.  On  A  ~<  x  /\  |^| { x  e.  On  |  A  ~<  x }  ~~  y )  ->  A  ~<  y )
21 breq2 4043 . . . . . . . . . . . . . 14  |-  ( x  =  y  ->  ( A  ~<  x  <->  A  ~<  y ) )
2221elrab 2936 . . . . . . . . . . . . 13  |-  ( y  e.  { x  e.  On  |  A  ~<  x }  <->  ( y  e.  On  /\  A  ~<  y ) )
23 ssrab2 3271 . . . . . . . . . . . . . 14  |-  { x  e.  On  |  A  ~<  x }  C_  On
24 onnmin 4610 . . . . . . . . . . . . . 14  |-  ( ( { x  e.  On  |  A  ~<  x }  C_  On  /\  y  e. 
{ x  e.  On  |  A  ~<  x }
)  ->  -.  y  e.  |^| { x  e.  On  |  A  ~<  x } )
2523, 24mpan 651 . . . . . . . . . . . . 13  |-  ( y  e.  { x  e.  On  |  A  ~<  x }  ->  -.  y  e.  |^| { x  e.  On  |  A  ~<  x } )
2622, 25sylbir 204 . . . . . . . . . . . 12  |-  ( ( y  e.  On  /\  A  ~<  y )  ->  -.  y  e.  |^| { x  e.  On  |  A  ~<  x } )
2726expcom 424 . . . . . . . . . . 11  |-  ( A 
~<  y  ->  ( y  e.  On  ->  -.  y  e.  |^| { x  e.  On  |  A  ~<  x } ) )
2820, 27syl 15 . . . . . . . . . 10  |-  ( ( E. x  e.  On  A  ~<  x  /\  |^| { x  e.  On  |  A  ~<  x }  ~~  y )  ->  (
y  e.  On  ->  -.  y  e.  |^| { x  e.  On  |  A  ~<  x } ) )
2928impancom 427 . . . . . . . . 9  |-  ( ( E. x  e.  On  A  ~<  x  /\  y  e.  On )  ->  ( |^| { x  e.  On  |  A  ~<  x }  ~~  y  ->  -.  y  e.  |^| { x  e.  On  |  A  ~<  x } ) )
3029con2d 107 . . . . . . . 8  |-  ( ( E. x  e.  On  A  ~<  x  /\  y  e.  On )  ->  (
y  e.  |^| { x  e.  On  |  A  ~<  x }  ->  -.  |^| { x  e.  On  |  A  ~<  x }  ~~  y ) )
3130impancom 427 . . . . . . 7  |-  ( ( E. x  e.  On  A  ~<  x  /\  y  e.  |^| { x  e.  On  |  A  ~<  x } )  ->  (
y  e.  On  ->  -. 
|^| { x  e.  On  |  A  ~<  x }  ~~  y ) )
3211, 31mpd 14 . . . . . 6  |-  ( ( E. x  e.  On  A  ~<  x  /\  y  e.  |^| { x  e.  On  |  A  ~<  x } )  ->  -.  |^|
{ x  e.  On  |  A  ~<  x }  ~~  y )
33 ensym 6926 . . . . . 6  |-  ( y 
~~  |^| { x  e.  On  |  A  ~<  x }  ->  |^| { x  e.  On  |  A  ~<  x }  ~~  y )
3432, 33nsyl 113 . . . . 5  |-  ( ( E. x  e.  On  A  ~<  x  /\  y  e.  |^| { x  e.  On  |  A  ~<  x } )  ->  -.  y  ~~  |^| { x  e.  On  |  A  ~<  x } )
35 brsdom 6900 . . . . 5  |-  ( y 
~<  |^| { x  e.  On  |  A  ~<  x }  <->  ( y  ~<_  |^|
{ x  e.  On  |  A  ~<  x }  /\  -.  y  ~~  |^| { x  e.  On  |  A  ~<  x } ) )
369, 34, 35sylanbrc 645 . . . 4  |-  ( ( E. x  e.  On  A  ~<  x  /\  y  e.  |^| { x  e.  On  |  A  ~<  x } )  ->  y  ~<  |^| { x  e.  On  |  A  ~<  x } )
3736ralrimiva 2639 . . 3  |-  ( E. x  e.  On  A  ~<  x  ->  A. y  e.  |^| { x  e.  On  |  A  ~<  x } y  ~<  |^| { x  e.  On  |  A  ~<  x } )
38 iscard 7624 . . 3  |-  ( (
card `  |^| { x  e.  On  |  A  ~<  x } )  =  |^| { x  e.  On  |  A  ~<  x }  <->  ( |^| { x  e.  On  |  A  ~<  x }  e.  On  /\  A. y  e. 
|^| { x  e.  On  |  A  ~<  x }
y  ~<  |^| { x  e.  On  |  A  ~<  x } ) )
392, 37, 38sylanbrc 645 . 2  |-  ( E. x  e.  On  A  ~<  x  ->  ( card ` 
|^| { x  e.  On  |  A  ~<  x }
)  =  |^| { x  e.  On  |  A  ~<  x } )
40 vprc 4168 . . . . . 6  |-  -.  _V  e.  _V
41 inteq 3881 . . . . . . . 8  |-  ( { x  e.  On  |  A  ~<  x }  =  (/) 
->  |^| { x  e.  On  |  A  ~<  x }  =  |^| (/) )
42 int0 3892 . . . . . . . 8  |-  |^| (/)  =  _V
4341, 42syl6eq 2344 . . . . . . 7  |-  ( { x  e.  On  |  A  ~<  x }  =  (/) 
->  |^| { x  e.  On  |  A  ~<  x }  =  _V )
4443eleq1d 2362 . . . . . 6  |-  ( { x  e.  On  |  A  ~<  x }  =  (/) 
->  ( |^| { x  e.  On  |  A  ~<  x }  e.  _V  <->  _V  e.  _V ) )
4540, 44mtbiri 294 . . . . 5  |-  ( { x  e.  On  |  A  ~<  x }  =  (/) 
->  -.  |^| { x  e.  On  |  A  ~<  x }  e.  _V )
46 fvex 5555 . . . . . 6  |-  ( card `  |^| { x  e.  On  |  A  ~<  x } )  e.  _V
47 eleq1 2356 . . . . . 6  |-  ( (
card `  |^| { x  e.  On  |  A  ~<  x } )  =  |^| { x  e.  On  |  A  ~<  x }  ->  ( ( card `  |^| { x  e.  On  |  A  ~<  x } )  e.  _V  <->  |^| { x  e.  On  |  A  ~<  x }  e.  _V )
)
4846, 47mpbii 202 . . . . 5  |-  ( (
card `  |^| { x  e.  On  |  A  ~<  x } )  =  |^| { x  e.  On  |  A  ~<  x }  ->  |^|
{ x  e.  On  |  A  ~<  x }  e.  _V )
4945, 48nsyl 113 . . . 4  |-  ( { x  e.  On  |  A  ~<  x }  =  (/) 
->  -.  ( card `  |^| { x  e.  On  |  A  ~<  x } )  =  |^| { x  e.  On  |  A  ~<  x } )
5049necon2ai 2504 . . 3  |-  ( (
card `  |^| { x  e.  On  |  A  ~<  x } )  =  |^| { x  e.  On  |  A  ~<  x }  ->  { x  e.  On  |  A  ~<  x }  =/=  (/) )
51 rabn0 3487 . . 3  |-  ( { x  e.  On  |  A  ~<  x }  =/=  (/)  <->  E. x  e.  On  A  ~<  x )
5250, 51sylib 188 . 2  |-  ( (
card `  |^| { x  e.  On  |  A  ~<  x } )  =  |^| { x  e.  On  |  A  ~<  x }  ->  E. x  e.  On  A  ~<  x )
5339, 52impbii 180 1  |-  ( E. x  e.  On  A  ~<  x  <->  ( card `  |^| { x  e.  On  |  A  ~<  x } )  =  |^| { x  e.  On  |  A  ~<  x } )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   E.wrex 2557   {crab 2560   _Vcvv 2801    C_ wss 3165   (/)c0 3468   |^|cint 3878   class class class wbr 4039   Ord word 4407   Oncon0 4408   ` cfv 5271    ~~ cen 6876    ~<_ cdom 6877    ~< csdm 6878   cardccrd 7584
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-int 3879  df-br 4040  df-opab 4094  df-mpt 4095  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-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-card 7588
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