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Theorem cardmin 8439
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 8374 . . 3  |-  ( A  e.  V  ->  E. x  e.  On  A  ~<  x
)
2 onintrab2 4782 . . 3  |-  ( E. x  e.  On  A  ~<  x  <->  |^| { x  e.  On  |  A  ~<  x }  e.  On )
31, 2sylib 189 . 2  |-  ( A  e.  V  ->  |^| { x  e.  On  |  A  ~<  x }  e.  On )
4 onelon 4606 . . . . . . . . 9  |-  ( (
|^| { x  e.  On  |  A  ~<  x }  e.  On  /\  y  e. 
|^| { x  e.  On  |  A  ~<  x }
)  ->  y  e.  On )
54ex 424 . . . . . . . 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 4216 . . . . . . . 8  |-  ( x  =  y  ->  ( A  ~<  x  <->  A  ~<  y ) )
87onnminsb 4784 . . . . . . 7  |-  ( y  e.  On  ->  (
y  e.  |^| { x  e.  On  |  A  ~<  x }  ->  -.  A  ~<  y ) )
96, 8syli 35 . . . . . 6  |-  ( A  e.  V  ->  (
y  e.  |^| { x  e.  On  |  A  ~<  x }  ->  -.  A  ~<  y ) )
10 vex 2959 . . . . . . 7  |-  y  e. 
_V
11 domtri 8431 . . . . . . 7  |-  ( ( y  e.  _V  /\  A  e.  V )  ->  ( y  ~<_  A  <->  -.  A  ~<  y ) )
1210, 11mpan 652 . . . . . 6  |-  ( A  e.  V  ->  (
y  ~<_  A  <->  -.  A  ~<  y ) )
139, 12sylibrd 226 . . . . 5  |-  ( A  e.  V  ->  (
y  e.  |^| { x  e.  On  |  A  ~<  x }  ->  y  ~<_  A ) )
14 nfcv 2572 . . . . . . . 8  |-  F/_ x A
15 nfcv 2572 . . . . . . . 8  |-  F/_ x  ~<
16 nfrab1 2888 . . . . . . . . 9  |-  F/_ x { x  e.  On  |  A  ~<  x }
1716nfint 4060 . . . . . . . 8  |-  F/_ x |^| { x  e.  On  |  A  ~<  x }
1814, 15, 17nfbr 4256 . . . . . . 7  |-  F/ x  A  ~<  |^| { x  e.  On  |  A  ~<  x }
19 breq2 4216 . . . . . . 7  |-  ( x  =  |^| { x  e.  On  |  A  ~<  x }  ->  ( A  ~<  x  <->  A  ~<  |^| { x  e.  On  |  A  ~<  x } ) )
2018, 19onminsb 4779 . . . . . 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 529 . . . 4  |-  ( A  e.  V  ->  (
y  e.  |^| { x  e.  On  |  A  ~<  x }  ->  ( y  ~<_  A  /\  A  ~<  |^| { x  e.  On  |  A  ~<  x } ) ) )
23 domsdomtr 7242 . . . 4  |-  ( ( y  ~<_  A  /\  A  ~<  |^| { x  e.  On  |  A  ~<  x } )  ->  y  ~<  |^| { x  e.  On  |  A  ~<  x } )
2422, 23syl6 31 . . 3  |-  ( A  e.  V  ->  (
y  e.  |^| { x  e.  On  |  A  ~<  x }  ->  y  ~<  |^|
{ x  e.  On  |  A  ~<  x }
) )
2524ralrimiv 2788 . 2  |-  ( A  e.  V  ->  A. y  e.  |^| { x  e.  On  |  A  ~<  x } y  ~<  |^| { x  e.  On  |  A  ~<  x } )
26 iscard 7862 . 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 646 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 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2705   E.wrex 2706   {crab 2709   _Vcvv 2956   |^|cint 4050   class class class wbr 4212   Oncon0 4581   ` cfv 5454    ~<_ cdom 7107    ~< csdm 7108   cardccrd 7822
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-ac2 8343
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-suc 4587  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-riota 6549  df-recs 6633  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-card 7826  df-ac 7997
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