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Theorem harval2 7630
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 7276 . . . . . . 7  |-  ( A  e.  dom  card  ->  (har
`  A )  =  { y  e.  On  |  y  ~<_  A }
)
21adantr 451 . . . . . 6  |-  ( ( A  e.  dom  card  /\  ( x  e.  On  /\  A  ~<  x )
)  ->  (har `  A
)  =  { y  e.  On  |  y  ~<_  A } )
3 domsdomtr 6996 . . . . . . . . . . . . 13  |-  ( ( y  ~<_  A  /\  A  ~<  x )  ->  y  ~<  x )
4 sdomel 7008 . . . . . . . . . . . . 13  |-  ( ( y  e.  On  /\  x  e.  On )  ->  ( y  ~<  x  ->  y  e.  x ) )
53, 4syl5 28 . . . . . . . . . . . 12  |-  ( ( y  e.  On  /\  x  e.  On )  ->  ( ( y  ~<_  A  /\  A  ~<  x
)  ->  y  e.  x ) )
65imp 418 . . . . . . . . . . 11  |-  ( ( ( y  e.  On  /\  x  e.  On )  /\  ( y  ~<_  A  /\  A  ~<  x
) )  ->  y  e.  x )
76an4s 799 . . . . . . . . . 10  |-  ( ( ( y  e.  On  /\  y  ~<_  A )  /\  ( x  e.  On  /\  A  ~<  x )
)  ->  y  e.  x )
87ancoms 439 . . . . . . . . 9  |-  ( ( ( x  e.  On  /\  A  ~<  x )  /\  ( y  e.  On  /\  y  ~<_  A ) )  ->  y  e.  x
)
983impb 1147 . . . . . . . 8  |-  ( ( ( x  e.  On  /\  A  ~<  x )  /\  y  e.  On  /\  y  ~<_  A )  -> 
y  e.  x )
109rabssdv 3253 . . . . . . 7  |-  ( ( x  e.  On  /\  A  ~<  x )  ->  { y  e.  On  |  y  ~<_  A }  C_  x )
1110adantl 452 . . . . . 6  |-  ( ( A  e.  dom  card  /\  ( x  e.  On  /\  A  ~<  x )
)  ->  { y  e.  On  |  y  ~<_  A }  C_  x )
122, 11eqsstrd 3212 . . . . 5  |-  ( ( A  e.  dom  card  /\  ( x  e.  On  /\  A  ~<  x )
)  ->  (har `  A
)  C_  x )
1312expr 598 . . . 4  |-  ( ( A  e.  dom  card  /\  x  e.  On )  ->  ( A  ~<  x  ->  (har `  A
)  C_  x )
)
1413ralrimiva 2626 . . 3  |-  ( A  e.  dom  card  ->  A. x  e.  On  ( A  ~<  x  ->  (har `  A )  C_  x
) )
15 ssintrab 3885 . . 3  |-  ( (har
`  A )  C_  |^|
{ x  e.  On  |  A  ~<  x }  <->  A. x  e.  On  ( A  ~<  x  ->  (har `  A )  C_  x
) )
1614, 15sylibr 203 . 2  |-  ( A  e.  dom  card  ->  (har
`  A )  C_  |^|
{ x  e.  On  |  A  ~<  x }
)
17 harcl 7275 . . . . 5  |-  (har `  A )  e.  On
1817a1i 10 . . . 4  |-  ( A  e.  dom  card  ->  (har
`  A )  e.  On )
19 harsdom 7628 . . . 4  |-  ( A  e.  dom  card  ->  A 
~<  (har `  A )
)
20 breq2 4027 . . . . 5  |-  ( x  =  (har `  A
)  ->  ( A  ~<  x  <->  A  ~<  (har `  A ) ) )
2120elrab 2923 . . . 4  |-  ( (har
`  A )  e. 
{ x  e.  On  |  A  ~<  x }  <->  ( (har `  A )  e.  On  /\  A  ~<  (har
`  A ) ) )
2218, 19, 21sylanbrc 645 . . 3  |-  ( A  e.  dom  card  ->  (har
`  A )  e. 
{ x  e.  On  |  A  ~<  x }
)
23 intss1 3877 . . 3  |-  ( (har
`  A )  e. 
{ x  e.  On  |  A  ~<  x }  ->  |^| { x  e.  On  |  A  ~<  x }  C_  (har `  A
) )
2422, 23syl 15 . 2  |-  ( A  e.  dom  card  ->  |^|
{ x  e.  On  |  A  ~<  x }  C_  (har `  A )
)
2516, 24eqssd 3196 1  |-  ( A  e.  dom  card  ->  (har
`  A )  = 
|^| { x  e.  On  |  A  ~<  x }
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   {crab 2547    C_ wss 3152   |^|cint 3862   class class class wbr 4023   Oncon0 4392   dom cdm 4689   ` cfv 5255    ~<_ cdom 6861    ~< csdm 6862  harchar 7270   cardccrd 7568
This theorem is referenced by:  alephnbtwn  7698
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-riota 6304  df-recs 6388  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-oi 7225  df-har 7272  df-card 7572
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