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Theorem cardmin2 7818
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 4722 . . . 4  |-  ( E. x  e.  On  A  ~<  x  <->  |^| { x  e.  On  |  A  ~<  x }  e.  On )
21biimpi 187 . . 3  |-  ( E. x  e.  On  A  ~<  x  ->  |^| { x  e.  On  |  A  ~<  x }  e.  On )
32adantr 452 . . . . . 6  |-  ( ( E. x  e.  On  A  ~<  x  /\  y  e.  |^| { x  e.  On  |  A  ~<  x } )  ->  |^| { x  e.  On  |  A  ~<  x }  e.  On )
4 eloni 4532 . . . . . . . 8  |-  ( |^| { x  e.  On  |  A  ~<  x }  e.  On  ->  Ord  |^| { x  e.  On  |  A  ~<  x } )
5 ordelss 4538 . . . . . . . 8  |-  ( ( Ord  |^| { x  e.  On  |  A  ~<  x }  /\  y  e. 
|^| { x  e.  On  |  A  ~<  x }
)  ->  y  C_  |^|
{ x  e.  On  |  A  ~<  x }
)
64, 5sylan 458 . . . . . . 7  |-  ( (
|^| { x  e.  On  |  A  ~<  x }  e.  On  /\  y  e. 
|^| { x  e.  On  |  A  ~<  x }
)  ->  y  C_  |^|
{ x  e.  On  |  A  ~<  x }
)
71, 6sylanb 459 . . . . . 6  |-  ( ( E. x  e.  On  A  ~<  x  /\  y  e.  |^| { x  e.  On  |  A  ~<  x } )  ->  y  C_ 
|^| { x  e.  On  |  A  ~<  x }
)
8 ssdomg 7089 . . . . . 6  |-  ( |^| { x  e.  On  |  A  ~<  x }  e.  On  ->  ( y  C_  |^|
{ x  e.  On  |  A  ~<  x }  ->  y  ~<_  |^| { x  e.  On  |  A  ~<  x } ) )
93, 7, 8sylc 58 . . . . 5  |-  ( ( E. x  e.  On  A  ~<  x  /\  y  e.  |^| { x  e.  On  |  A  ~<  x } )  ->  y  ~<_  |^|
{ x  e.  On  |  A  ~<  x }
)
10 onelon 4547 . . . . . . . 8  |-  ( (
|^| { x  e.  On  |  A  ~<  x }  e.  On  /\  y  e. 
|^| { x  e.  On  |  A  ~<  x }
)  ->  y  e.  On )
111, 10sylanb 459 . . . . . . 7  |-  ( ( E. x  e.  On  A  ~<  x  /\  y  e.  |^| { x  e.  On  |  A  ~<  x } )  ->  y  e.  On )
12 nfcv 2523 . . . . . . . . . . . . . 14  |-  F/_ x A
13 nfcv 2523 . . . . . . . . . . . . . 14  |-  F/_ x  ~<
14 nfrab1 2831 . . . . . . . . . . . . . . 15  |-  F/_ x { x  e.  On  |  A  ~<  x }
1514nfint 4002 . . . . . . . . . . . . . 14  |-  F/_ x |^| { x  e.  On  |  A  ~<  x }
1612, 13, 15nfbr 4197 . . . . . . . . . . . . 13  |-  F/ x  A  ~<  |^| { x  e.  On  |  A  ~<  x }
17 breq2 4157 . . . . . . . . . . . . 13  |-  ( x  =  |^| { x  e.  On  |  A  ~<  x }  ->  ( A  ~<  x  <->  A  ~<  |^| { x  e.  On  |  A  ~<  x } ) )
1816, 17onminsb 4719 . . . . . . . . . . . 12  |-  ( E. x  e.  On  A  ~<  x  ->  A  ~<  |^|
{ x  e.  On  |  A  ~<  x }
)
19 sdomentr 7177 . . . . . . . . . . . 12  |-  ( ( A  ~<  |^| { x  e.  On  |  A  ~<  x }  /\  |^| { x  e.  On  |  A  ~<  x }  ~~  y )  ->  A  ~<  y
)
2018, 19sylan 458 . . . . . . . . . . 11  |-  ( ( E. x  e.  On  A  ~<  x  /\  |^| { x  e.  On  |  A  ~<  x }  ~~  y )  ->  A  ~<  y )
21 breq2 4157 . . . . . . . . . . . . . 14  |-  ( x  =  y  ->  ( A  ~<  x  <->  A  ~<  y ) )
2221elrab 3035 . . . . . . . . . . . . 13  |-  ( y  e.  { x  e.  On  |  A  ~<  x }  <->  ( y  e.  On  /\  A  ~<  y ) )
23 ssrab2 3371 . . . . . . . . . . . . . 14  |-  { x  e.  On  |  A  ~<  x }  C_  On
24 onnmin 4723 . . . . . . . . . . . . . 14  |-  ( ( { x  e.  On  |  A  ~<  x }  C_  On  /\  y  e. 
{ x  e.  On  |  A  ~<  x }
)  ->  -.  y  e.  |^| { x  e.  On  |  A  ~<  x } )
2523, 24mpan 652 . . . . . . . . . . . . 13  |-  ( y  e.  { x  e.  On  |  A  ~<  x }  ->  -.  y  e.  |^| { x  e.  On  |  A  ~<  x } )
2622, 25sylbir 205 . . . . . . . . . . . 12  |-  ( ( y  e.  On  /\  A  ~<  y )  ->  -.  y  e.  |^| { x  e.  On  |  A  ~<  x } )
2726expcom 425 . . . . . . . . . . 11  |-  ( A 
~<  y  ->  ( y  e.  On  ->  -.  y  e.  |^| { x  e.  On  |  A  ~<  x } ) )
2820, 27syl 16 . . . . . . . . . 10  |-  ( ( E. x  e.  On  A  ~<  x  /\  |^| { x  e.  On  |  A  ~<  x }  ~~  y )  ->  (
y  e.  On  ->  -.  y  e.  |^| { x  e.  On  |  A  ~<  x } ) )
2928impancom 428 . . . . . . . . 9  |-  ( ( E. x  e.  On  A  ~<  x  /\  y  e.  On )  ->  ( |^| { x  e.  On  |  A  ~<  x }  ~~  y  ->  -.  y  e.  |^| { x  e.  On  |  A  ~<  x } ) )
3029con2d 109 . . . . . . . 8  |-  ( ( E. x  e.  On  A  ~<  x  /\  y  e.  On )  ->  (
y  e.  |^| { x  e.  On  |  A  ~<  x }  ->  -.  |^| { x  e.  On  |  A  ~<  x }  ~~  y ) )
3130impancom 428 . . . . . . 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 15 . . . . . 6  |-  ( ( E. x  e.  On  A  ~<  x  /\  y  e.  |^| { x  e.  On  |  A  ~<  x } )  ->  -.  |^|
{ x  e.  On  |  A  ~<  x }  ~~  y )
33 ensym 7092 . . . . . 6  |-  ( y 
~~  |^| { x  e.  On  |  A  ~<  x }  ->  |^| { x  e.  On  |  A  ~<  x }  ~~  y )
3432, 33nsyl 115 . . . . 5  |-  ( ( E. x  e.  On  A  ~<  x  /\  y  e.  |^| { x  e.  On  |  A  ~<  x } )  ->  -.  y  ~~  |^| { x  e.  On  |  A  ~<  x } )
35 brsdom 7066 . . . . 5  |-  ( y 
~<  |^| { x  e.  On  |  A  ~<  x }  <->  ( y  ~<_  |^|
{ x  e.  On  |  A  ~<  x }  /\  -.  y  ~~  |^| { x  e.  On  |  A  ~<  x } ) )
369, 34, 35sylanbrc 646 . . . 4  |-  ( ( E. x  e.  On  A  ~<  x  /\  y  e.  |^| { x  e.  On  |  A  ~<  x } )  ->  y  ~<  |^| { x  e.  On  |  A  ~<  x } )
3736ralrimiva 2732 . . 3  |-  ( E. x  e.  On  A  ~<  x  ->  A. y  e.  |^| { x  e.  On  |  A  ~<  x } y  ~<  |^| { x  e.  On  |  A  ~<  x } )
38 iscard 7795 . . 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 646 . 2  |-  ( E. x  e.  On  A  ~<  x  ->  ( card ` 
|^| { x  e.  On  |  A  ~<  x }
)  =  |^| { x  e.  On  |  A  ~<  x } )
40 vprc 4282 . . . . . 6  |-  -.  _V  e.  _V
41 inteq 3995 . . . . . . . 8  |-  ( { x  e.  On  |  A  ~<  x }  =  (/) 
->  |^| { x  e.  On  |  A  ~<  x }  =  |^| (/) )
42 int0 4006 . . . . . . . 8  |-  |^| (/)  =  _V
4341, 42syl6eq 2435 . . . . . . 7  |-  ( { x  e.  On  |  A  ~<  x }  =  (/) 
->  |^| { x  e.  On  |  A  ~<  x }  =  _V )
4443eleq1d 2453 . . . . . 6  |-  ( { x  e.  On  |  A  ~<  x }  =  (/) 
->  ( |^| { x  e.  On  |  A  ~<  x }  e.  _V  <->  _V  e.  _V ) )
4540, 44mtbiri 295 . . . . 5  |-  ( { x  e.  On  |  A  ~<  x }  =  (/) 
->  -.  |^| { x  e.  On  |  A  ~<  x }  e.  _V )
46 fvex 5682 . . . . . 6  |-  ( card `  |^| { x  e.  On  |  A  ~<  x } )  e.  _V
47 eleq1 2447 . . . . . 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 203 . . . . 5  |-  ( (
card `  |^| { x  e.  On  |  A  ~<  x } )  =  |^| { x  e.  On  |  A  ~<  x }  ->  |^|
{ x  e.  On  |  A  ~<  x }  e.  _V )
4945, 48nsyl 115 . . . 4  |-  ( { x  e.  On  |  A  ~<  x }  =  (/) 
->  -.  ( card `  |^| { x  e.  On  |  A  ~<  x } )  =  |^| { x  e.  On  |  A  ~<  x } )
5049necon2ai 2595 . . 3  |-  ( (
card `  |^| { x  e.  On  |  A  ~<  x } )  =  |^| { x  e.  On  |  A  ~<  x }  ->  { x  e.  On  |  A  ~<  x }  =/=  (/) )
51 rabn0 3590 . . 3  |-  ( { x  e.  On  |  A  ~<  x }  =/=  (/)  <->  E. x  e.  On  A  ~<  x )
5250, 51sylib 189 . 2  |-  ( (
card `  |^| { x  e.  On  |  A  ~<  x } )  =  |^| { x  e.  On  |  A  ~<  x }  ->  E. x  e.  On  A  ~<  x )
5339, 52impbii 181 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 177    /\ wa 359    = wceq 1649    e. wcel 1717    =/= wne 2550   A.wral 2649   E.wrex 2650   {crab 2653   _Vcvv 2899    C_ wss 3263   (/)c0 3571   |^|cint 3992   class class class wbr 4153   Ord word 4521   Oncon0 4522   ` cfv 5394    ~~ cen 7042    ~<_ cdom 7043    ~< csdm 7044   cardccrd 7755
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-er 6841  df-en 7046  df-dom 7047  df-sdom 7048  df-card 7759
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