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Theorem harval2 8381
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 7991 . . . . . . 7  |-  ( A  e.  dom  card  ->  (har
`  A )  =  { y  e.  On  |  y  ~<_  A }
)
21adantr 465 . . . . . 6  |-  ( ( A  e.  dom  card  /\  ( x  e.  On  /\  A  ~<  x )
)  ->  (har `  A
)  =  { y  e.  On  |  y  ~<_  A } )
3 domsdomtr 7654 . . . . . . . . . . . . 13  |-  ( ( y  ~<_  A  /\  A  ~<  x )  ->  y  ~<  x )
4 sdomel 7666 . . . . . . . . . . . . 13  |-  ( ( y  e.  On  /\  x  e.  On )  ->  ( y  ~<  x  ->  y  e.  x ) )
53, 4syl5 32 . . . . . . . . . . . 12  |-  ( ( y  e.  On  /\  x  e.  On )  ->  ( ( y  ~<_  A  /\  A  ~<  x
)  ->  y  e.  x ) )
65imp 429 . . . . . . . . . . 11  |-  ( ( ( y  e.  On  /\  x  e.  On )  /\  ( y  ~<_  A  /\  A  ~<  x
) )  ->  y  e.  x )
76an4s 826 . . . . . . . . . 10  |-  ( ( ( y  e.  On  /\  y  ~<_  A )  /\  ( x  e.  On  /\  A  ~<  x )
)  ->  y  e.  x )
87ancoms 453 . . . . . . . . 9  |-  ( ( ( x  e.  On  /\  A  ~<  x )  /\  ( y  e.  On  /\  y  ~<_  A ) )  ->  y  e.  x
)
983impb 1193 . . . . . . . 8  |-  ( ( ( x  e.  On  /\  A  ~<  x )  /\  y  e.  On  /\  y  ~<_  A )  -> 
y  e.  x )
109rabssdv 3565 . . . . . . 7  |-  ( ( x  e.  On  /\  A  ~<  x )  ->  { y  e.  On  |  y  ~<_  A }  C_  x )
1110adantl 466 . . . . . 6  |-  ( ( A  e.  dom  card  /\  ( x  e.  On  /\  A  ~<  x )
)  ->  { y  e.  On  |  y  ~<_  A }  C_  x )
122, 11eqsstrd 3523 . . . . 5  |-  ( ( A  e.  dom  card  /\  ( x  e.  On  /\  A  ~<  x )
)  ->  (har `  A
)  C_  x )
1312expr 615 . . . 4  |-  ( ( A  e.  dom  card  /\  x  e.  On )  ->  ( A  ~<  x  ->  (har `  A
)  C_  x )
)
1413ralrimiva 2857 . . 3  |-  ( A  e.  dom  card  ->  A. x  e.  On  ( A  ~<  x  ->  (har `  A )  C_  x
) )
15 ssintrab 4295 . . 3  |-  ( (har
`  A )  C_  |^|
{ x  e.  On  |  A  ~<  x }  <->  A. x  e.  On  ( A  ~<  x  ->  (har `  A )  C_  x
) )
1614, 15sylibr 212 . 2  |-  ( A  e.  dom  card  ->  (har
`  A )  C_  |^|
{ x  e.  On  |  A  ~<  x }
)
17 harcl 7990 . . . . 5  |-  (har `  A )  e.  On
1817a1i 11 . . . 4  |-  ( A  e.  dom  card  ->  (har
`  A )  e.  On )
19 harsdom 8379 . . . 4  |-  ( A  e.  dom  card  ->  A 
~<  (har `  A )
)
20 breq2 4441 . . . . 5  |-  ( x  =  (har `  A
)  ->  ( A  ~<  x  <->  A  ~<  (har `  A ) ) )
2120elrab 3243 . . . 4  |-  ( (har
`  A )  e. 
{ x  e.  On  |  A  ~<  x }  <->  ( (har `  A )  e.  On  /\  A  ~<  (har
`  A ) ) )
2218, 19, 21sylanbrc 664 . . 3  |-  ( A  e.  dom  card  ->  (har
`  A )  e. 
{ x  e.  On  |  A  ~<  x }
)
23 intss1 4286 . . 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 3506 1  |-  ( A  e.  dom  card  ->  (har
`  A )  = 
|^| { x  e.  On  |  A  ~<  x }
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1383    e. wcel 1804   A.wral 2793   {crab 2797    C_ wss 3461   |^|cint 4271   class class class wbr 4437   Oncon0 4868   dom cdm 4989   ` cfv 5578    ~<_ cdom 7516    ~< csdm 7517  harchar 7985   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
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-lim 4873  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-oi 7938  df-har 7987  df-card 8323
This theorem is referenced by:  alephnbtwn  8455
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