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Theorem cardmin 8231
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 8166 . . 3  |-  ( A  e.  V  ->  E. x  e.  On  A  ~<  x
)
2 onintrab2 4630 . . 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 4454 . . . . . . . . 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 4064 . . . . . . . 8  |-  ( x  =  y  ->  ( A  ~<  x  <->  A  ~<  y ) )
87onnminsb 4632 . . . . . . 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 2825 . . . . . . 7  |-  y  e. 
_V
11 domtri 8223 . . . . . . 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 2452 . . . . . . . 8  |-  F/_ x A
15 nfcv 2452 . . . . . . . 8  |-  F/_ x  ~<
16 nfrab1 2754 . . . . . . . . 9  |-  F/_ x { x  e.  On  |  A  ~<  x }
1716nfint 3909 . . . . . . . 8  |-  F/_ x |^| { x  e.  On  |  A  ~<  x }
1814, 15, 17nfbr 4104 . . . . . . 7  |-  F/ x  A  ~<  |^| { x  e.  On  |  A  ~<  x }
19 breq2 4064 . . . . . . 7  |-  ( x  =  |^| { x  e.  On  |  A  ~<  x }  ->  ( A  ~<  x  <->  A  ~<  |^| { x  e.  On  |  A  ~<  x } ) )
2018, 19onminsb 4627 . . . . . 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 7039 . . . 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 2659 . 2  |-  ( A  e.  V  ->  A. y  e.  |^| { x  e.  On  |  A  ~<  x } y  ~<  |^| { x  e.  On  |  A  ~<  x } )
26 iscard 7653 . 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 1633    e. wcel 1701   A.wral 2577   E.wrex 2578   {crab 2581   _Vcvv 2822   |^|cint 3899   class class class wbr 4060   Oncon0 4429   ` cfv 5292    ~<_ cdom 6904    ~< csdm 6905   cardccrd 7613
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-ac2 8134
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 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-int 3900  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-se 4390  df-we 4391  df-ord 4432  df-on 4433  df-suc 4435  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-isom 5301  df-riota 6346  df-recs 6430  df-er 6702  df-en 6907  df-dom 6908  df-sdom 6909  df-card 7617  df-ac 7788
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