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Theorem cardmin 8831
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 8766 . . 3  |-  ( A  e.  V  ->  E. x  e.  On  A  ~<  x
)
2 onintrab2 6515 . . 3  |-  ( E. x  e.  On  A  ~<  x  <->  |^| { x  e.  On  |  A  ~<  x }  e.  On )
31, 2sylib 196 . 2  |-  ( A  e.  V  ->  |^| { x  e.  On  |  A  ~<  x }  e.  On )
4 onelon 4844 . . . . . . . . 9  |-  ( (
|^| { x  e.  On  |  A  ~<  x }  e.  On  /\  y  e. 
|^| { x  e.  On  |  A  ~<  x }
)  ->  y  e.  On )
54ex 434 . . . . . . . 8  |-  ( |^| { x  e.  On  |  A  ~<  x }  e.  On  ->  ( y  e. 
|^| { x  e.  On  |  A  ~<  x }  ->  y  e.  On ) )
63, 5syl 16 . . . . . . 7  |-  ( A  e.  V  ->  (
y  e.  |^| { x  e.  On  |  A  ~<  x }  ->  y  e.  On ) )
7 breq2 4396 . . . . . . . 8  |-  ( x  =  y  ->  ( A  ~<  x  <->  A  ~<  y ) )
87onnminsb 6517 . . . . . . 7  |-  ( y  e.  On  ->  (
y  e.  |^| { x  e.  On  |  A  ~<  x }  ->  -.  A  ~<  y ) )
96, 8syli 37 . . . . . 6  |-  ( A  e.  V  ->  (
y  e.  |^| { x  e.  On  |  A  ~<  x }  ->  -.  A  ~<  y ) )
10 vex 3073 . . . . . . 7  |-  y  e. 
_V
11 domtri 8823 . . . . . . 7  |-  ( ( y  e.  _V  /\  A  e.  V )  ->  ( y  ~<_  A  <->  -.  A  ~<  y ) )
1210, 11mpan 670 . . . . . 6  |-  ( A  e.  V  ->  (
y  ~<_  A  <->  -.  A  ~<  y ) )
139, 12sylibrd 234 . . . . 5  |-  ( A  e.  V  ->  (
y  e.  |^| { x  e.  On  |  A  ~<  x }  ->  y  ~<_  A ) )
14 nfcv 2613 . . . . . . . 8  |-  F/_ x A
15 nfcv 2613 . . . . . . . 8  |-  F/_ x  ~<
16 nfrab1 2999 . . . . . . . . 9  |-  F/_ x { x  e.  On  |  A  ~<  x }
1716nfint 4238 . . . . . . . 8  |-  F/_ x |^| { x  e.  On  |  A  ~<  x }
1814, 15, 17nfbr 4436 . . . . . . 7  |-  F/ x  A  ~<  |^| { x  e.  On  |  A  ~<  x }
19 breq2 4396 . . . . . . 7  |-  ( x  =  |^| { x  e.  On  |  A  ~<  x }  ->  ( A  ~<  x  <->  A  ~<  |^| { x  e.  On  |  A  ~<  x } ) )
2018, 19onminsb 6512 . . . . . 6  |-  ( E. x  e.  On  A  ~<  x  ->  A  ~<  |^|
{ x  e.  On  |  A  ~<  x }
)
211, 20syl 16 . . . . 5  |-  ( A  e.  V  ->  A  ~<  |^| { x  e.  On  |  A  ~<  x } )
2213, 21jctird 544 . . . 4  |-  ( A  e.  V  ->  (
y  e.  |^| { x  e.  On  |  A  ~<  x }  ->  ( y  ~<_  A  /\  A  ~<  |^| { x  e.  On  |  A  ~<  x } ) ) )
23 domsdomtr 7548 . . . 4  |-  ( ( y  ~<_  A  /\  A  ~<  |^| { x  e.  On  |  A  ~<  x } )  ->  y  ~<  |^| { x  e.  On  |  A  ~<  x } )
2422, 23syl6 33 . . 3  |-  ( A  e.  V  ->  (
y  e.  |^| { x  e.  On  |  A  ~<  x }  ->  y  ~<  |^|
{ x  e.  On  |  A  ~<  x }
) )
2524ralrimiv 2820 . 2  |-  ( A  e.  V  ->  A. y  e.  |^| { x  e.  On  |  A  ~<  x } y  ~<  |^| { x  e.  On  |  A  ~<  x } )
26 iscard 8248 . 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 664 1  |-  ( A  e.  V  ->  ( 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   A.wral 2795   E.wrex 2796   {crab 2799   _Vcvv 3070   |^|cint 4228   class class class wbr 4392   Oncon0 4819   ` cfv 5518    ~<_ cdom 7410    ~< csdm 7411   cardccrd 8208
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 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-ac2 8735
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 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-int 4229  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-eprel 4732  df-id 4736  df-po 4741  df-so 4742  df-fr 4779  df-se 4780  df-we 4781  df-ord 4822  df-on 4823  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-isom 5527  df-riota 6153  df-recs 6934  df-er 7203  df-en 7413  df-dom 7414  df-sdom 7415  df-card 8212  df-ac 8389
This theorem is referenced by: (None)
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