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Theorem cardmin 8942
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 8877 . . 3  |-  ( A  e.  V  ->  E. x  e.  On  A  ~<  x
)
2 onintrab2 6622 . . 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 4893 . . . . . . . . 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 4441 . . . . . . . 8  |-  ( x  =  y  ->  ( A  ~<  x  <->  A  ~<  y ) )
87onnminsb 6624 . . . . . . 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 3098 . . . . . . 7  |-  y  e. 
_V
11 domtri 8934 . . . . . . 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 2605 . . . . . . . 8  |-  F/_ x A
15 nfcv 2605 . . . . . . . 8  |-  F/_ x  ~<
16 nfrab1 3024 . . . . . . . . 9  |-  F/_ x { x  e.  On  |  A  ~<  x }
1716nfint 4281 . . . . . . . 8  |-  F/_ x |^| { x  e.  On  |  A  ~<  x }
1814, 15, 17nfbr 4481 . . . . . . 7  |-  F/ x  A  ~<  |^| { x  e.  On  |  A  ~<  x }
19 breq2 4441 . . . . . . 7  |-  ( x  =  |^| { x  e.  On  |  A  ~<  x }  ->  ( A  ~<  x  <->  A  ~<  |^| { x  e.  On  |  A  ~<  x } ) )
2018, 19onminsb 6619 . . . . . 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 7654 . . . 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 2855 . 2  |-  ( A  e.  V  ->  A. y  e.  |^| { x  e.  On  |  A  ~<  x } y  ~<  |^| { x  e.  On  |  A  ~<  x } )
26 iscard 8359 . 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 1383    e. wcel 1804   A.wral 2793   E.wrex 2794   {crab 2797   _Vcvv 3095   |^|cint 4271   class class class wbr 4437   Oncon0 4868   ` cfv 5578    ~<_ cdom 7516    ~< csdm 7517   cardccrd 8319
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-ac2 8846
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-se 4829  df-we 4830  df-ord 4871  df-on 4872  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-riota 6242  df-recs 7044  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-card 8323  df-ac 8500
This theorem is referenced by: (None)
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