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Theorem harval2 7646
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 7292 . . . . . . 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 7012 . . . . . . . . . . . . 13  |-  ( ( y  ~<_  A  /\  A  ~<  x )  ->  y  ~<  x )
4 sdomel 7024 . . . . . . . . . . . . 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 3266 . . . . . . 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 3225 . . . . 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 2639 . . 3  |-  ( A  e.  dom  card  ->  A. x  e.  On  ( A  ~<  x  ->  (har `  A )  C_  x
) )
15 ssintrab 3901 . . 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 7291 . . . . 5  |-  (har `  A )  e.  On
1817a1i 10 . . . 4  |-  ( A  e.  dom  card  ->  (har
`  A )  e.  On )
19 harsdom 7644 . . . 4  |-  ( A  e.  dom  card  ->  A 
~<  (har `  A )
)
20 breq2 4043 . . . . 5  |-  ( x  =  (har `  A
)  ->  ( A  ~<  x  <->  A  ~<  (har `  A ) ) )
2120elrab 2936 . . . 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 3893 . . 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 3209 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 1632    e. wcel 1696   A.wral 2556   {crab 2560    C_ wss 3165   |^|cint 3878   class class class wbr 4039   Oncon0 4408   dom cdm 4705   ` cfv 5271    ~<_ cdom 6877    ~< csdm 6878  harchar 7286   cardccrd 7584
This theorem is referenced by:  alephnbtwn  7714
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-riota 6320  df-recs 6404  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-oi 7241  df-har 7288  df-card 7588
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