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Theorem cardmin 8186
Description: The smallest ordinal that strictly dominates a set is a cardinal. (Contributed by NM, 28-Oct-2003.) (Revised by Mario Carneiro, 20-Sep-2014.)
Assertion
Ref Expression
cardmin  |-  ( A  e.  V  ->  ( card `  |^| { x  e.  On  |  A  ~<  x } )  =  |^| { x  e.  On  |  A  ~<  x } )
Distinct variable group:    x, A
Allowed substitution hint:    V( x)

Proof of Theorem cardmin
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 numthcor 8121 . . 3  |-  ( A  e.  V  ->  E. x  e.  On  A  ~<  x
)
2 onintrab2 4593 . . 3  |-  ( E. x  e.  On  A  ~<  x  <->  |^| { x  e.  On  |  A  ~<  x }  e.  On )
31, 2sylib 188 . 2  |-  ( A  e.  V  ->  |^| { x  e.  On  |  A  ~<  x }  e.  On )
4 onelon 4417 . . . . . . . . 9  |-  ( (
|^| { x  e.  On  |  A  ~<  x }  e.  On  /\  y  e. 
|^| { x  e.  On  |  A  ~<  x }
)  ->  y  e.  On )
54ex 423 . . . . . . . 8  |-  ( |^| { x  e.  On  |  A  ~<  x }  e.  On  ->  ( y  e. 
|^| { x  e.  On  |  A  ~<  x }  ->  y  e.  On ) )
63, 5syl 15 . . . . . . 7  |-  ( A  e.  V  ->  (
y  e.  |^| { x  e.  On  |  A  ~<  x }  ->  y  e.  On ) )
7 breq2 4027 . . . . . . . 8  |-  ( x  =  y  ->  ( A  ~<  x  <->  A  ~<  y ) )
87onnminsb 4595 . . . . . . 7  |-  ( y  e.  On  ->  (
y  e.  |^| { x  e.  On  |  A  ~<  x }  ->  -.  A  ~<  y ) )
96, 8syli 33 . . . . . 6  |-  ( A  e.  V  ->  (
y  e.  |^| { x  e.  On  |  A  ~<  x }  ->  -.  A  ~<  y ) )
10 vex 2791 . . . . . . 7  |-  y  e. 
_V
11 domtri 8178 . . . . . . 7  |-  ( ( y  e.  _V  /\  A  e.  V )  ->  ( y  ~<_  A  <->  -.  A  ~<  y ) )
1210, 11mpan 651 . . . . . 6  |-  ( A  e.  V  ->  (
y  ~<_  A  <->  -.  A  ~<  y ) )
139, 12sylibrd 225 . . . . 5  |-  ( A  e.  V  ->  (
y  e.  |^| { x  e.  On  |  A  ~<  x }  ->  y  ~<_  A ) )
14 nfcv 2419 . . . . . . . 8  |-  F/_ x A
15 nfcv 2419 . . . . . . . 8  |-  F/_ x  ~<
16 nfrab1 2720 . . . . . . . . 9  |-  F/_ x { x  e.  On  |  A  ~<  x }
1716nfint 3872 . . . . . . . 8  |-  F/_ x |^| { x  e.  On  |  A  ~<  x }
1814, 15, 17nfbr 4067 . . . . . . 7  |-  F/ x  A  ~<  |^| { x  e.  On  |  A  ~<  x }
19 breq2 4027 . . . . . . 7  |-  ( x  =  |^| { x  e.  On  |  A  ~<  x }  ->  ( A  ~<  x  <->  A  ~<  |^| { x  e.  On  |  A  ~<  x } ) )
2018, 19onminsb 4590 . . . . . 6  |-  ( E. x  e.  On  A  ~<  x  ->  A  ~<  |^|
{ x  e.  On  |  A  ~<  x }
)
211, 20syl 15 . . . . 5  |-  ( A  e.  V  ->  A  ~<  |^| { x  e.  On  |  A  ~<  x } )
2213, 21jctird 528 . . . 4  |-  ( A  e.  V  ->  (
y  e.  |^| { x  e.  On  |  A  ~<  x }  ->  ( y  ~<_  A  /\  A  ~<  |^| { x  e.  On  |  A  ~<  x } ) ) )
23 domsdomtr 6996 . . . 4  |-  ( ( y  ~<_  A  /\  A  ~<  |^| { x  e.  On  |  A  ~<  x } )  ->  y  ~<  |^| { x  e.  On  |  A  ~<  x } )
2422, 23syl6 29 . . 3  |-  ( A  e.  V  ->  (
y  e.  |^| { x  e.  On  |  A  ~<  x }  ->  y  ~<  |^|
{ x  e.  On  |  A  ~<  x }
) )
2524ralrimiv 2625 . 2  |-  ( A  e.  V  ->  A. y  e.  |^| { x  e.  On  |  A  ~<  x } y  ~<  |^| { x  e.  On  |  A  ~<  x } )
26 iscard 7608 . 2  |-  ( (
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 } ) )
273, 25, 26sylanbrc 645 1  |-  ( A  e.  V  ->  ( 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 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   {crab 2547   _Vcvv 2788   |^|cint 3862   class class class wbr 4023   Oncon0 4392   ` cfv 5255    ~<_ cdom 6861    ~< csdm 6862   cardccrd 7568
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-ac2 8089
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-suc 4398  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-riota 6304  df-recs 6388  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-card 7572  df-ac 7743
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