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Theorem nodenselem6 27964
Description: The restriction of a surreal to the abstraction from nodenselem4 27962 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 27962 . . . 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 27930 . . . . . 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 27963 . . . . . 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 27958 . . . . . . 7  |-  ( bday `  A )  e.  On
76onelssi 4928 . . . . . 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 5625 . . . . 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 5235 . . . . . 6  |-  ( A  |`  |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } )  C_  A
12 rnss 5169 . . . . . 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 27929 . . . . . 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 3468 . . . 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 5523 . . . 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 5644 . . . 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 3172 . . 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 27924 . 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 1370    e. wcel 1758    =/= wne 2644   E.wrex 2796   {crab 2799    C_ wss 3429   {cpr 3980   |^|cint 4229   class class class wbr 4393   Oncon0 4820   ran crn 4942    |` cres 4943    Fn wfn 5514   -->wf 5515   ` cfv 5519   1oc1o 7016   2oc2o 7017   Nocsur 27918   <scslt 27919   bdaycbday 27920
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-int 4230  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-we 4782  df-ord 4823  df-on 4824  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-1o 7023  df-2o 7024  df-no 27921  df-slt 27922  df-bday 27923
This theorem is referenced by:  nodense  27967
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