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Theorem nodenselem6 30146
Description: The restriction of a surreal to the abstraction from nodenselem4 30144 is still a surreal. (Contributed by Scott Fenton, 16-Jun-2011.)
Assertion
Ref Expression
nodenselem6  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( ( bday `  A )  =  (
bday `  B )  /\  A <s B ) )  ->  ( A  |`  |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } )  e.  No )
Distinct variable groups:    A, a    B, a

Proof of Theorem nodenselem6
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 nodenselem4 30144 . . . 4  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  A <s
B )  ->  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  On )
21adantrl 714 . . 3  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( ( bday `  A )  =  (
bday `  B )  /\  A <s B ) )  ->  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  On )
3 nofnbday 30112 . . . . . 6  |-  ( A  e.  No  ->  A  Fn  ( bday `  A
) )
43ad2antrr 724 . . . . 5  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( ( bday `  A )  =  (
bday `  B )  /\  A <s B ) )  ->  A  Fn  ( bday `  A
) )
5 nodenselem5 30145 . . . . . 6  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( ( bday `  A )  =  (
bday `  B )  /\  A <s B ) )  ->  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  ( bday `  A )
)
6 bdayelon 30140 . . . . . . 7  |-  ( bday `  A )  e.  On
76onelssi 5518 . . . . . 6  |-  ( |^| { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) }  e.  ( bday `  A
)  ->  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  C_  ( bday `  A )
)
85, 7syl 17 . . . . 5  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( ( bday `  A )  =  (
bday `  B )  /\  A <s B ) )  ->  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  C_  ( bday `  A )
)
9 fnssres 5675 . . . . 5  |-  ( ( A  Fn  ( bday `  A )  /\  |^| { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) } 
C_  ( bday `  A
) )  ->  ( A  |`  |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } )  Fn 
|^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )
104, 8, 9syl2anc 659 . . . 4  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( ( bday `  A )  =  (
bday `  B )  /\  A <s B ) )  ->  ( A  |`  |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } )  Fn 
|^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )
11 resss 5117 . . . . . 6  |-  ( A  |`  |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } )  C_  A
12 rnss 5052 . . . . . 6  |-  ( ( A  |`  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) 
C_  A  ->  ran  ( A  |`  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) 
C_  ran  A )
1311, 12ax-mp 5 . . . . 5  |-  ran  ( A  |`  |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } )  C_  ran  A
14 norn 30111 . . . . . 6  |-  ( A  e.  No  ->  ran  A 
C_  { 1o ,  2o } )
1514ad2antrr 724 . . . . 5  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( ( bday `  A )  =  (
bday `  B )  /\  A <s B ) )  ->  ran  A 
C_  { 1o ,  2o } )
1613, 15syl5ss 3453 . . . 4  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( ( bday `  A )  =  (
bday `  B )  /\  A <s B ) )  ->  ran  ( A  |`  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) 
C_  { 1o ,  2o } )
17 df-f 5573 . . . 4  |-  ( ( A  |`  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) : |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } --> { 1o ,  2o }  <->  ( ( A  |`  |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } )  Fn 
|^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  /\  ran  ( A  |`  |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } )  C_  { 1o ,  2o }
) )
1810, 16, 17sylanbrc 662 . . 3  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( ( bday `  A )  =  (
bday `  B )  /\  A <s B ) )  ->  ( A  |`  |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } ) :
|^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }
--> { 1o ,  2o } )
19 feq2 5697 . . . 4  |-  ( x  =  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  ->  ( ( A  |`  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) : x --> { 1o ,  2o }  <->  ( A  |` 
|^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) : |^| { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) } --> { 1o ,  2o } ) )
2019rspcev 3160 . . 3  |-  ( (
|^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  On  /\  ( A  |`  |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } ) :
|^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }
--> { 1o ,  2o } )  ->  E. x  e.  On  ( A  |`  |^|
{ a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) : x --> { 1o ,  2o } )
212, 18, 20syl2anc 659 . 2  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( ( bday `  A )  =  (
bday `  B )  /\  A <s B ) )  ->  E. x  e.  On  ( A  |`  |^|
{ a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) : x --> { 1o ,  2o } )
22 elno 30106 . 2  |-  ( ( A  |`  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  e.  No  <->  E. x  e.  On  ( A  |`  |^|
{ a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) : x --> { 1o ,  2o } )
2321, 22sylibr 212 1  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( ( bday `  A )  =  (
bday `  B )  /\  A <s B ) )  ->  ( A  |`  |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } )  e.  No )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842    =/= wne 2598   E.wrex 2755   {crab 2758    C_ wss 3414   {cpr 3974   |^|cint 4227   class class class wbr 4395   ran crn 4824    |` cres 4825   Oncon0 5410    Fn wfn 5564   -->wf 5565   ` cfv 5569   1oc1o 7160   2oc2o 7161   Nocsur 30100   <scslt 30101   bdaycbday 30102
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-ord 5413  df-on 5414  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-1o 7167  df-2o 7168  df-no 30103  df-slt 30104  df-bday 30105
This theorem is referenced by:  nodense  30149
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