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Theorem cardmin2 8379
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 6621 . . . 4  |-  ( E. x  e.  On  A  ~<  x  <->  |^| { x  e.  On  |  A  ~<  x }  e.  On )
21biimpi 194 . . 3  |-  ( E. x  e.  On  A  ~<  x  ->  |^| { x  e.  On  |  A  ~<  x }  e.  On )
32adantr 465 . . . . . 6  |-  ( ( E. x  e.  On  A  ~<  x  /\  y  e.  |^| { x  e.  On  |  A  ~<  x } )  ->  |^| { x  e.  On  |  A  ~<  x }  e.  On )
4 eloni 4888 . . . . . . . 8  |-  ( |^| { x  e.  On  |  A  ~<  x }  e.  On  ->  Ord  |^| { x  e.  On  |  A  ~<  x } )
5 ordelss 4894 . . . . . . . 8  |-  ( ( Ord  |^| { x  e.  On  |  A  ~<  x }  /\  y  e. 
|^| { x  e.  On  |  A  ~<  x }
)  ->  y  C_  |^|
{ x  e.  On  |  A  ~<  x }
)
64, 5sylan 471 . . . . . . 7  |-  ( (
|^| { x  e.  On  |  A  ~<  x }  e.  On  /\  y  e. 
|^| { x  e.  On  |  A  ~<  x }
)  ->  y  C_  |^|
{ x  e.  On  |  A  ~<  x }
)
71, 6sylanb 472 . . . . . 6  |-  ( ( E. x  e.  On  A  ~<  x  /\  y  e.  |^| { x  e.  On  |  A  ~<  x } )  ->  y  C_ 
|^| { x  e.  On  |  A  ~<  x }
)
8 ssdomg 7561 . . . . . 6  |-  ( |^| { x  e.  On  |  A  ~<  x }  e.  On  ->  ( y  C_  |^|
{ x  e.  On  |  A  ~<  x }  ->  y  ~<_  |^| { x  e.  On  |  A  ~<  x } ) )
93, 7, 8sylc 60 . . . . 5  |-  ( ( E. x  e.  On  A  ~<  x  /\  y  e.  |^| { x  e.  On  |  A  ~<  x } )  ->  y  ~<_  |^|
{ x  e.  On  |  A  ~<  x }
)
10 onelon 4903 . . . . . . . 8  |-  ( (
|^| { x  e.  On  |  A  ~<  x }  e.  On  /\  y  e. 
|^| { x  e.  On  |  A  ~<  x }
)  ->  y  e.  On )
111, 10sylanb 472 . . . . . . 7  |-  ( ( E. x  e.  On  A  ~<  x  /\  y  e.  |^| { x  e.  On  |  A  ~<  x } )  ->  y  e.  On )
12 nfcv 2629 . . . . . . . . . . . . . 14  |-  F/_ x A
13 nfcv 2629 . . . . . . . . . . . . . 14  |-  F/_ x  ~<
14 nfrab1 3042 . . . . . . . . . . . . . . 15  |-  F/_ x { x  e.  On  |  A  ~<  x }
1514nfint 4292 . . . . . . . . . . . . . 14  |-  F/_ x |^| { x  e.  On  |  A  ~<  x }
1612, 13, 15nfbr 4491 . . . . . . . . . . . . 13  |-  F/ x  A  ~<  |^| { x  e.  On  |  A  ~<  x }
17 breq2 4451 . . . . . . . . . . . . 13  |-  ( x  =  |^| { x  e.  On  |  A  ~<  x }  ->  ( A  ~<  x  <->  A  ~<  |^| { x  e.  On  |  A  ~<  x } ) )
1816, 17onminsb 6618 . . . . . . . . . . . 12  |-  ( E. x  e.  On  A  ~<  x  ->  A  ~<  |^|
{ x  e.  On  |  A  ~<  x }
)
19 sdomentr 7651 . . . . . . . . . . . 12  |-  ( ( A  ~<  |^| { x  e.  On  |  A  ~<  x }  /\  |^| { x  e.  On  |  A  ~<  x }  ~~  y )  ->  A  ~<  y
)
2018, 19sylan 471 . . . . . . . . . . 11  |-  ( ( E. x  e.  On  A  ~<  x  /\  |^| { x  e.  On  |  A  ~<  x }  ~~  y )  ->  A  ~<  y )
21 breq2 4451 . . . . . . . . . . . . . 14  |-  ( x  =  y  ->  ( A  ~<  x  <->  A  ~<  y ) )
2221elrab 3261 . . . . . . . . . . . . 13  |-  ( y  e.  { x  e.  On  |  A  ~<  x }  <->  ( y  e.  On  /\  A  ~<  y ) )
23 ssrab2 3585 . . . . . . . . . . . . . 14  |-  { x  e.  On  |  A  ~<  x }  C_  On
24 onnmin 6622 . . . . . . . . . . . . . 14  |-  ( ( { x  e.  On  |  A  ~<  x }  C_  On  /\  y  e. 
{ x  e.  On  |  A  ~<  x }
)  ->  -.  y  e.  |^| { x  e.  On  |  A  ~<  x } )
2523, 24mpan 670 . . . . . . . . . . . . 13  |-  ( y  e.  { x  e.  On  |  A  ~<  x }  ->  -.  y  e.  |^| { x  e.  On  |  A  ~<  x } )
2622, 25sylbir 213 . . . . . . . . . . . 12  |-  ( ( y  e.  On  /\  A  ~<  y )  ->  -.  y  e.  |^| { x  e.  On  |  A  ~<  x } )
2726expcom 435 . . . . . . . . . . 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 440 . . . . . . . . 9  |-  ( ( E. x  e.  On  A  ~<  x  /\  y  e.  On )  ->  ( |^| { x  e.  On  |  A  ~<  x }  ~~  y  ->  -.  y  e.  |^| { x  e.  On  |  A  ~<  x } ) )
3029con2d 115 . . . . . . . 8  |-  ( ( E. x  e.  On  A  ~<  x  /\  y  e.  On )  ->  (
y  e.  |^| { x  e.  On  |  A  ~<  x }  ->  -.  |^| { x  e.  On  |  A  ~<  x }  ~~  y ) )
3130impancom 440 . . . . . . 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 7564 . . . . . 6  |-  ( y 
~~  |^| { x  e.  On  |  A  ~<  x }  ->  |^| { x  e.  On  |  A  ~<  x }  ~~  y )
3432, 33nsyl 121 . . . . 5  |-  ( ( E. x  e.  On  A  ~<  x  /\  y  e.  |^| { x  e.  On  |  A  ~<  x } )  ->  -.  y  ~~  |^| { x  e.  On  |  A  ~<  x } )
35 brsdom 7538 . . . . 5  |-  ( y 
~<  |^| { x  e.  On  |  A  ~<  x }  <->  ( y  ~<_  |^|
{ x  e.  On  |  A  ~<  x }  /\  -.  y  ~~  |^| { x  e.  On  |  A  ~<  x } ) )
369, 34, 35sylanbrc 664 . . . 4  |-  ( ( E. x  e.  On  A  ~<  x  /\  y  e.  |^| { x  e.  On  |  A  ~<  x } )  ->  y  ~<  |^| { x  e.  On  |  A  ~<  x } )
3736ralrimiva 2878 . . 3  |-  ( E. x  e.  On  A  ~<  x  ->  A. y  e.  |^| { x  e.  On  |  A  ~<  x } y  ~<  |^| { x  e.  On  |  A  ~<  x } )
38 iscard 8356 . . 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 664 . 2  |-  ( E. x  e.  On  A  ~<  x  ->  ( card ` 
|^| { x  e.  On  |  A  ~<  x }
)  =  |^| { x  e.  On  |  A  ~<  x } )
40 vprc 4585 . . . . . 6  |-  -.  _V  e.  _V
41 inteq 4285 . . . . . . . 8  |-  ( { x  e.  On  |  A  ~<  x }  =  (/) 
->  |^| { x  e.  On  |  A  ~<  x }  =  |^| (/) )
42 int0 4296 . . . . . . . 8  |-  |^| (/)  =  _V
4341, 42syl6eq 2524 . . . . . . 7  |-  ( { x  e.  On  |  A  ~<  x }  =  (/) 
->  |^| { x  e.  On  |  A  ~<  x }  =  _V )
4443eleq1d 2536 . . . . . 6  |-  ( { x  e.  On  |  A  ~<  x }  =  (/) 
->  ( |^| { x  e.  On  |  A  ~<  x }  e.  _V  <->  _V  e.  _V ) )
4540, 44mtbiri 303 . . . . 5  |-  ( { x  e.  On  |  A  ~<  x }  =  (/) 
->  -.  |^| { x  e.  On  |  A  ~<  x }  e.  _V )
46 fvex 5876 . . . . . 6  |-  ( card `  |^| { x  e.  On  |  A  ~<  x } )  e.  _V
47 eleq1 2539 . . . . . 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 211 . . . . 5  |-  ( (
card `  |^| { x  e.  On  |  A  ~<  x } )  =  |^| { x  e.  On  |  A  ~<  x }  ->  |^|
{ x  e.  On  |  A  ~<  x }  e.  _V )
4945, 48nsyl 121 . . . 4  |-  ( { x  e.  On  |  A  ~<  x }  =  (/) 
->  -.  ( card `  |^| { x  e.  On  |  A  ~<  x } )  =  |^| { x  e.  On  |  A  ~<  x } )
5049necon2ai 2702 . . 3  |-  ( (
card `  |^| { x  e.  On  |  A  ~<  x } )  =  |^| { x  e.  On  |  A  ~<  x }  ->  { x  e.  On  |  A  ~<  x }  =/=  (/) )
51 rabn0 3805 . . 3  |-  ( { x  e.  On  |  A  ~<  x }  =/=  (/)  <->  E. x  e.  On  A  ~<  x )
5250, 51sylib 196 . 2  |-  ( (
card `  |^| { x  e.  On  |  A  ~<  x } )  =  |^| { x  e.  On  |  A  ~<  x }  ->  E. x  e.  On  A  ~<  x )
5339, 52impbii 188 1  |-  ( E. x  e.  On  A  ~<  x  <->  ( card `  |^| { x  e.  On  |  A  ~<  x } )  =  |^| { x  e.  On  |  A  ~<  x } )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   E.wrex 2815   {crab 2818   _Vcvv 3113    C_ wss 3476   (/)c0 3785   |^|cint 4282   class class class wbr 4447   Ord word 4877   Oncon0 4878   ` cfv 5588    ~~ cen 7513    ~<_ cdom 7514    ~< csdm 7515   cardccrd 8316
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-card 8320
This theorem is referenced by: (None)
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