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Theorem cardmin2 8274
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 6518 . . . 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 4832 . . . . . . . 8  |-  ( |^| { x  e.  On  |  A  ~<  x }  e.  On  ->  Ord  |^| { x  e.  On  |  A  ~<  x } )
5 ordelss 4838 . . . . . . . 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 7460 . . . . . 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 4847 . . . . . . . 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 2614 . . . . . . . . . . . . . 14  |-  F/_ x A
13 nfcv 2614 . . . . . . . . . . . . . 14  |-  F/_ x  ~<
14 nfrab1 3001 . . . . . . . . . . . . . . 15  |-  F/_ x { x  e.  On  |  A  ~<  x }
1514nfint 4241 . . . . . . . . . . . . . 14  |-  F/_ x |^| { x  e.  On  |  A  ~<  x }
1612, 13, 15nfbr 4439 . . . . . . . . . . . . 13  |-  F/ x  A  ~<  |^| { x  e.  On  |  A  ~<  x }
17 breq2 4399 . . . . . . . . . . . . 13  |-  ( x  =  |^| { x  e.  On  |  A  ~<  x }  ->  ( A  ~<  x  <->  A  ~<  |^| { x  e.  On  |  A  ~<  x } ) )
1816, 17onminsb 6515 . . . . . . . . . . . 12  |-  ( E. x  e.  On  A  ~<  x  ->  A  ~<  |^|
{ x  e.  On  |  A  ~<  x }
)
19 sdomentr 7550 . . . . . . . . . . . 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 4399 . . . . . . . . . . . . . 14  |-  ( x  =  y  ->  ( A  ~<  x  <->  A  ~<  y ) )
2221elrab 3218 . . . . . . . . . . . . 13  |-  ( y  e.  { x  e.  On  |  A  ~<  x }  <->  ( y  e.  On  /\  A  ~<  y ) )
23 ssrab2 3540 . . . . . . . . . . . . . 14  |-  { x  e.  On  |  A  ~<  x }  C_  On
24 onnmin 6519 . . . . . . . . . . . . . 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 7463 . . . . . 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 7437 . . . . 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 2827 . . 3  |-  ( E. x  e.  On  A  ~<  x  ->  A. y  e.  |^| { x  e.  On  |  A  ~<  x } y  ~<  |^| { x  e.  On  |  A  ~<  x } )
38 iscard 8251 . . 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 4533 . . . . . 6  |-  -.  _V  e.  _V
41 inteq 4234 . . . . . . . 8  |-  ( { x  e.  On  |  A  ~<  x }  =  (/) 
->  |^| { x  e.  On  |  A  ~<  x }  =  |^| (/) )
42 int0 4245 . . . . . . . 8  |-  |^| (/)  =  _V
4341, 42syl6eq 2509 . . . . . . 7  |-  ( { x  e.  On  |  A  ~<  x }  =  (/) 
->  |^| { x  e.  On  |  A  ~<  x }  =  _V )
4443eleq1d 2521 . . . . . 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 5804 . . . . . 6  |-  ( card `  |^| { x  e.  On  |  A  ~<  x } )  e.  _V
47 eleq1 2524 . . . . . 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 2684 . . 3  |-  ( (
card `  |^| { x  e.  On  |  A  ~<  x } )  =  |^| { x  e.  On  |  A  ~<  x }  ->  { x  e.  On  |  A  ~<  x }  =/=  (/) )
51 rabn0 3760 . . 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 1370    e. wcel 1758    =/= wne 2645   A.wral 2796   E.wrex 2797   {crab 2800   _Vcvv 3072    C_ wss 3431   (/)c0 3740   |^|cint 4231   class class class wbr 4395   Ord word 4821   Oncon0 4822   ` cfv 5521    ~~ cen 7412    ~<_ cdom 7413    ~< csdm 7414   cardccrd 8211
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-sbc 3289  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-int 4232  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-ord 4825  df-on 4826  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-er 7206  df-en 7416  df-dom 7417  df-sdom 7418  df-card 8215
This theorem is referenced by: (None)
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