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Theorem harval2 7810
Description: An alternative expression for the Hartogs number of a well-orderable set. (Contributed by Mario Carneiro, 15-May-2015.)
Assertion
Ref Expression
harval2  |-  ( A  e.  dom  card  ->  (har
`  A )  = 
|^| { x  e.  On  |  A  ~<  x }
)
Distinct variable group:    x, A

Proof of Theorem harval2
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 harval 7456 . . . . . . 7  |-  ( A  e.  dom  card  ->  (har
`  A )  =  { y  e.  On  |  y  ~<_  A }
)
21adantr 452 . . . . . 6  |-  ( ( A  e.  dom  card  /\  ( x  e.  On  /\  A  ~<  x )
)  ->  (har `  A
)  =  { y  e.  On  |  y  ~<_  A } )
3 domsdomtr 7171 . . . . . . . . . . . . 13  |-  ( ( y  ~<_  A  /\  A  ~<  x )  ->  y  ~<  x )
4 sdomel 7183 . . . . . . . . . . . . 13  |-  ( ( y  e.  On  /\  x  e.  On )  ->  ( y  ~<  x  ->  y  e.  x ) )
53, 4syl5 30 . . . . . . . . . . . 12  |-  ( ( y  e.  On  /\  x  e.  On )  ->  ( ( y  ~<_  A  /\  A  ~<  x
)  ->  y  e.  x ) )
65imp 419 . . . . . . . . . . 11  |-  ( ( ( y  e.  On  /\  x  e.  On )  /\  ( y  ~<_  A  /\  A  ~<  x
) )  ->  y  e.  x )
76an4s 800 . . . . . . . . . 10  |-  ( ( ( y  e.  On  /\  y  ~<_  A )  /\  ( x  e.  On  /\  A  ~<  x )
)  ->  y  e.  x )
87ancoms 440 . . . . . . . . 9  |-  ( ( ( x  e.  On  /\  A  ~<  x )  /\  ( y  e.  On  /\  y  ~<_  A ) )  ->  y  e.  x
)
983impb 1149 . . . . . . . 8  |-  ( ( ( x  e.  On  /\  A  ~<  x )  /\  y  e.  On  /\  y  ~<_  A )  -> 
y  e.  x )
109rabssdv 3359 . . . . . . 7  |-  ( ( x  e.  On  /\  A  ~<  x )  ->  { y  e.  On  |  y  ~<_  A }  C_  x )
1110adantl 453 . . . . . 6  |-  ( ( A  e.  dom  card  /\  ( x  e.  On  /\  A  ~<  x )
)  ->  { y  e.  On  |  y  ~<_  A }  C_  x )
122, 11eqsstrd 3318 . . . . 5  |-  ( ( A  e.  dom  card  /\  ( x  e.  On  /\  A  ~<  x )
)  ->  (har `  A
)  C_  x )
1312expr 599 . . . 4  |-  ( ( A  e.  dom  card  /\  x  e.  On )  ->  ( A  ~<  x  ->  (har `  A
)  C_  x )
)
1413ralrimiva 2725 . . 3  |-  ( A  e.  dom  card  ->  A. x  e.  On  ( A  ~<  x  ->  (har `  A )  C_  x
) )
15 ssintrab 4008 . . 3  |-  ( (har
`  A )  C_  |^|
{ x  e.  On  |  A  ~<  x }  <->  A. x  e.  On  ( A  ~<  x  ->  (har `  A )  C_  x
) )
1614, 15sylibr 204 . 2  |-  ( A  e.  dom  card  ->  (har
`  A )  C_  |^|
{ x  e.  On  |  A  ~<  x }
)
17 harcl 7455 . . . . 5  |-  (har `  A )  e.  On
1817a1i 11 . . . 4  |-  ( A  e.  dom  card  ->  (har
`  A )  e.  On )
19 harsdom 7808 . . . 4  |-  ( A  e.  dom  card  ->  A 
~<  (har `  A )
)
20 breq2 4150 . . . . 5  |-  ( x  =  (har `  A
)  ->  ( A  ~<  x  <->  A  ~<  (har `  A ) ) )
2120elrab 3028 . . . 4  |-  ( (har
`  A )  e. 
{ x  e.  On  |  A  ~<  x }  <->  ( (har `  A )  e.  On  /\  A  ~<  (har
`  A ) ) )
2218, 19, 21sylanbrc 646 . . 3  |-  ( A  e.  dom  card  ->  (har
`  A )  e. 
{ x  e.  On  |  A  ~<  x }
)
23 intss1 4000 . . 3  |-  ( (har
`  A )  e. 
{ x  e.  On  |  A  ~<  x }  ->  |^| { x  e.  On  |  A  ~<  x }  C_  (har `  A
) )
2422, 23syl 16 . 2  |-  ( A  e.  dom  card  ->  |^|
{ x  e.  On  |  A  ~<  x }  C_  (har `  A )
)
2516, 24eqssd 3301 1  |-  ( A  e.  dom  card  ->  (har
`  A )  = 
|^| { x  e.  On  |  A  ~<  x }
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2642   {crab 2646    C_ wss 3256   |^|cint 3985   class class class wbr 4146   Oncon0 4515   dom cdm 4811   ` cfv 5387    ~<_ cdom 7036    ~< csdm 7037  harchar 7450   cardccrd 7748
This theorem is referenced by:  alephnbtwn  7878
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-int 3986  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-se 4476  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-isom 5396  df-riota 6478  df-recs 6562  df-er 6834  df-en 7039  df-dom 7040  df-sdom 7041  df-oi 7405  df-har 7452  df-card 7752
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