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Theorem nodenselem6 29009
Description: The restriction of a surreal to the abstraction from nodenselem4 29007 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 29007 . . . 4  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  A <s
B )  ->  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  On )
21adantrl 715 . . 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 28975 . . . . . 6  |-  ( A  e.  No  ->  A  Fn  ( bday `  A
) )
43ad2antrr 725 . . . . 5  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( ( bday `  A )  =  (
bday `  B )  /\  A <s B ) )  ->  A  Fn  ( bday `  A
) )
5 nodenselem5 29008 . . . . . 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 29003 . . . . . . 7  |-  ( bday `  A )  e.  On
76onelssi 4979 . . . . . 6  |-  ( |^| { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) }  e.  ( bday `  A
)  ->  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  C_  ( bday `  A )
)
85, 7syl 16 . . . . 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 5685 . . . . 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 661 . . . 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 5288 . . . . . 6  |-  ( A  |`  |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } )  C_  A
12 rnss 5222 . . . . . 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 28974 . . . . . 6  |-  ( A  e.  No  ->  ran  A 
C_  { 1o ,  2o } )
1514ad2antrr 725 . . . . 5  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( ( bday `  A )  =  (
bday `  B )  /\  A <s B ) )  ->  ran  A 
C_  { 1o ,  2o } )
1613, 15syl5ss 3508 . . . 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 5583 . . . 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 664 . . 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 5705 . . . 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 3207 . . 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 661 . 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 28969 . 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 369    = wceq 1374    e. wcel 1762    =/= wne 2655   E.wrex 2808   {crab 2811    C_ wss 3469   {cpr 4022   |^|cint 4275   class class class wbr 4440   Oncon0 4871   ran crn 4993    |` cres 4994    Fn wfn 5574   -->wf 5575   ` cfv 5579   1oc1o 7113   2oc2o 7114   Nocsur 28963   <scslt 28964   bdaycbday 28965
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-1o 7120  df-2o 7121  df-no 28966  df-slt 28967  df-bday 28968
This theorem is referenced by:  nodense  29012
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