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Theorem harval2 7876
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 7522 . . . . . . 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 7234 . . . . . . . . . . . . 13  |-  ( ( y  ~<_  A  /\  A  ~<  x )  ->  y  ~<  x )
4 sdomel 7246 . . . . . . . . . . . . 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 3415 . . . . . . 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 3374 . . . . 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 2781 . . 3  |-  ( A  e.  dom  card  ->  A. x  e.  On  ( A  ~<  x  ->  (har `  A )  C_  x
) )
15 ssintrab 4065 . . 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 7521 . . . . 5  |-  (har `  A )  e.  On
1817a1i 11 . . . 4  |-  ( A  e.  dom  card  ->  (har
`  A )  e.  On )
19 harsdom 7874 . . . 4  |-  ( A  e.  dom  card  ->  A 
~<  (har `  A )
)
20 breq2 4208 . . . . 5  |-  ( x  =  (har `  A
)  ->  ( A  ~<  x  <->  A  ~<  (har `  A ) ) )
2120elrab 3084 . . . 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 4057 . . 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 3357 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 1652    e. wcel 1725   A.wral 2697   {crab 2701    C_ wss 3312   |^|cint 4042   class class class wbr 4204   Oncon0 4573   dom cdm 4870   ` cfv 5446    ~<_ cdom 7099    ~< csdm 7100  harchar 7516   cardccrd 7814
This theorem is referenced by:  alephnbtwn  7944
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 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-riota 6541  df-recs 6625  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-oi 7471  df-har 7518  df-card 7818
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