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Theorem cardmin2 8411
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 6620 . . . 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 463 . . . . . 6  |-  ( ( E. x  e.  On  A  ~<  x  /\  y  e.  |^| { x  e.  On  |  A  ~<  x } )  ->  |^| { x  e.  On  |  A  ~<  x }  e.  On )
4 eloni 5420 . . . . . . . 8  |-  ( |^| { x  e.  On  |  A  ~<  x }  e.  On  ->  Ord  |^| { x  e.  On  |  A  ~<  x } )
5 ordelss 5426 . . . . . . . 8  |-  ( ( Ord  |^| { x  e.  On  |  A  ~<  x }  /\  y  e. 
|^| { x  e.  On  |  A  ~<  x }
)  ->  y  C_  |^|
{ x  e.  On  |  A  ~<  x }
)
64, 5sylan 469 . . . . . . 7  |-  ( (
|^| { x  e.  On  |  A  ~<  x }  e.  On  /\  y  e. 
|^| { x  e.  On  |  A  ~<  x }
)  ->  y  C_  |^|
{ x  e.  On  |  A  ~<  x }
)
71, 6sylanb 470 . . . . . 6  |-  ( ( E. x  e.  On  A  ~<  x  /\  y  e.  |^| { x  e.  On  |  A  ~<  x } )  ->  y  C_ 
|^| { x  e.  On  |  A  ~<  x }
)
8 ssdomg 7599 . . . . . 6  |-  ( |^| { x  e.  On  |  A  ~<  x }  e.  On  ->  ( y  C_  |^|
{ x  e.  On  |  A  ~<  x }  ->  y  ~<_  |^| { x  e.  On  |  A  ~<  x } ) )
93, 7, 8sylc 59 . . . . 5  |-  ( ( E. x  e.  On  A  ~<  x  /\  y  e.  |^| { x  e.  On  |  A  ~<  x } )  ->  y  ~<_  |^|
{ x  e.  On  |  A  ~<  x }
)
10 onelon 5435 . . . . . . . 8  |-  ( (
|^| { x  e.  On  |  A  ~<  x }  e.  On  /\  y  e. 
|^| { x  e.  On  |  A  ~<  x }
)  ->  y  e.  On )
111, 10sylanb 470 . . . . . . 7  |-  ( ( E. x  e.  On  A  ~<  x  /\  y  e.  |^| { x  e.  On  |  A  ~<  x } )  ->  y  e.  On )
12 nfcv 2564 . . . . . . . . . . . . . 14  |-  F/_ x A
13 nfcv 2564 . . . . . . . . . . . . . 14  |-  F/_ x  ~<
14 nfrab1 2988 . . . . . . . . . . . . . . 15  |-  F/_ x { x  e.  On  |  A  ~<  x }
1514nfint 4237 . . . . . . . . . . . . . 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 6617 . . . . . . . . . . . 12  |-  ( E. x  e.  On  A  ~<  x  ->  A  ~<  |^|
{ x  e.  On  |  A  ~<  x }
)
19 sdomentr 7689 . . . . . . . . . . . 12  |-  ( ( A  ~<  |^| { x  e.  On  |  A  ~<  x }  /\  |^| { x  e.  On  |  A  ~<  x }  ~~  y )  ->  A  ~<  y
)
2018, 19sylan 469 . . . . . . . . . . 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 3207 . . . . . . . . . . . . 13  |-  ( y  e.  { x  e.  On  |  A  ~<  x }  <->  ( y  e.  On  /\  A  ~<  y ) )
23 ssrab2 3524 . . . . . . . . . . . . . 14  |-  { x  e.  On  |  A  ~<  x }  C_  On
24 onnmin 6621 . . . . . . . . . . . . . 14  |-  ( ( { x  e.  On  |  A  ~<  x }  C_  On  /\  y  e. 
{ x  e.  On  |  A  ~<  x }
)  ->  -.  y  e.  |^| { x  e.  On  |  A  ~<  x } )
2523, 24mpan 668 . . . . . . . . . . . . 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 433 . . . . . . . . . . 11  |-  ( A 
~<  y  ->  ( y  e.  On  ->  -.  y  e.  |^| { x  e.  On  |  A  ~<  x } ) )
2820, 27syl 17 . . . . . . . . . 10  |-  ( ( E. x  e.  On  A  ~<  x  /\  |^| { x  e.  On  |  A  ~<  x }  ~~  y )  ->  (
y  e.  On  ->  -.  y  e.  |^| { x  e.  On  |  A  ~<  x } ) )
2928impancom 438 . . . . . . . . 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 438 . . . . . . 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 7602 . . . . . 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 7576 . . . . 5  |-  ( y 
~<  |^| { x  e.  On  |  A  ~<  x }  <->  ( y  ~<_  |^|
{ x  e.  On  |  A  ~<  x }  /\  -.  y  ~~  |^| { x  e.  On  |  A  ~<  x } ) )
369, 34, 35sylanbrc 662 . . . 4  |-  ( ( E. x  e.  On  A  ~<  x  /\  y  e.  |^| { x  e.  On  |  A  ~<  x } )  ->  y  ~<  |^| { x  e.  On  |  A  ~<  x } )
3736ralrimiva 2818 . . 3  |-  ( E. x  e.  On  A  ~<  x  ->  A. y  e.  |^| { x  e.  On  |  A  ~<  x } y  ~<  |^| { x  e.  On  |  A  ~<  x } )
38 iscard 8388 . . 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 662 . 2  |-  ( E. x  e.  On  A  ~<  x  ->  ( card ` 
|^| { x  e.  On  |  A  ~<  x }
)  =  |^| { x  e.  On  |  A  ~<  x } )
40 vprc 4532 . . . . . 6  |-  -.  _V  e.  _V
41 inteq 4230 . . . . . . . 8  |-  ( { x  e.  On  |  A  ~<  x }  =  (/) 
->  |^| { x  e.  On  |  A  ~<  x }  =  |^| (/) )
42 int0 4241 . . . . . . . 8  |-  |^| (/)  =  _V
4341, 42syl6eq 2459 . . . . . . 7  |-  ( { x  e.  On  |  A  ~<  x }  =  (/) 
->  |^| { x  e.  On  |  A  ~<  x }  =  _V )
4443eleq1d 2471 . . . . . 6  |-  ( { x  e.  On  |  A  ~<  x }  =  (/) 
->  ( |^| { x  e.  On  |  A  ~<  x }  e.  _V  <->  _V  e.  _V ) )
4540, 44mtbiri 301 . . . . 5  |-  ( { x  e.  On  |  A  ~<  x }  =  (/) 
->  -.  |^| { x  e.  On  |  A  ~<  x }  e.  _V )
46 fvex 5859 . . . . . 6  |-  ( card `  |^| { x  e.  On  |  A  ~<  x } )  e.  _V
47 eleq1 2474 . . . . . 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 2638 . . 3  |-  ( (
card `  |^| { x  e.  On  |  A  ~<  x } )  =  |^| { x  e.  On  |  A  ~<  x }  ->  { x  e.  On  |  A  ~<  x }  =/=  (/) )
51 rabn0 3759 . . 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 187 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 367    = wceq 1405    e. wcel 1842    =/= wne 2598   A.wral 2754   E.wrex 2755   {crab 2758   _Vcvv 3059    C_ wss 3414   (/)c0 3738   |^|cint 4227   class class class wbr 4395   Ord word 5409   Oncon0 5410   ` cfv 5569    ~~ cen 7551    ~<_ cdom 7552    ~< csdm 7553   cardccrd 8348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-sbc 3278  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-ord 5413  df-on 5414  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-er 7348  df-en 7555  df-dom 7556  df-sdom 7557  df-card 8352
This theorem is referenced by: (None)
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