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Theorem nodenselem5 27826
Description: Lemma for nodense 27830. If the birthdays of two distinct surreals are equal, then the ordinal from nodenselem4 27825 is an element of that birthday. (Contributed by Scott Fenton, 16-Jun-2011.)
Assertion
Ref Expression
nodenselem5  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( ( bday `  A )  =  (
bday `  B )  /\  A <s B ) )  ->  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  ( bday `  A )
)
Distinct variable groups:    A, a    B, a

Proof of Theorem nodenselem5
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 sltirr 27811 . . . . . . . . 9  |-  ( A  e.  No  ->  -.  A <s A )
2 breq2 4296 . . . . . . . . . 10  |-  ( A  =  B  ->  ( A <s A  <->  A <s B ) )
32notbid 294 . . . . . . . . 9  |-  ( A  =  B  ->  ( -.  A <s A  <->  -.  A <s B ) )
41, 3syl5ibcom 220 . . . . . . . 8  |-  ( A  e.  No  ->  ( A  =  B  ->  -.  A <s B ) )
54con2d 115 . . . . . . 7  |-  ( A  e.  No  ->  ( A <s B  ->  -.  A  =  B
) )
65imp 429 . . . . . 6  |-  ( ( A  e.  No  /\  A <s B )  ->  -.  A  =  B )
76ad2ant2rl 748 . . . . 5  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( dom  A  =  dom  B  /\  A <s B ) )  ->  -.  A  =  B )
8 nofun 27790 . . . . . . . . 9  |-  ( A  e.  No  ->  Fun  A )
9 nofun 27790 . . . . . . . . 9  |-  ( B  e.  No  ->  Fun  B )
10 eqfunfv 5802 . . . . . . . . 9  |-  ( ( Fun  A  /\  Fun  B )  ->  ( A  =  B  <->  ( dom  A  =  dom  B  /\  A. x  e.  dom  A ( A `  x )  =  ( B `  x ) ) ) )
118, 9, 10syl2an 477 . . . . . . . 8  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( A  =  B  <-> 
( dom  A  =  dom  B  /\  A. x  e.  dom  A ( A `
 x )  =  ( B `  x
) ) ) )
1211notbid 294 . . . . . . 7  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( -.  A  =  B  <->  -.  ( dom  A  =  dom  B  /\  A. x  e.  dom  A
( A `  x
)  =  ( B `
 x ) ) ) )
1312adantr 465 . . . . . 6  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( dom  A  =  dom  B  /\  A <s B ) )  ->  ( -.  A  =  B  <->  -.  ( dom  A  =  dom  B  /\  A. x  e.  dom  A
( A `  x
)  =  ( B `
 x ) ) ) )
14 imnan 422 . . . . . . . . . . . 12  |-  ( ( dom  A  =  dom  B  ->  -.  A. x  e.  dom  A ( A `
 x )  =  ( B `  x
) )  <->  -.  ( dom  A  =  dom  B  /\  A. x  e.  dom  A ( A `  x
)  =  ( B `
 x ) ) )
1514biimpri 206 . . . . . . . . . . 11  |-  ( -.  ( dom  A  =  dom  B  /\  A. x  e.  dom  A ( A `  x )  =  ( B `  x ) )  -> 
( dom  A  =  dom  B  ->  -.  A. x  e.  dom  A ( A `
 x )  =  ( B `  x
) ) )
1615impcom 430 . . . . . . . . . 10  |-  ( ( dom  A  =  dom  B  /\  -.  ( dom 
A  =  dom  B  /\  A. x  e.  dom  A ( A `  x
)  =  ( B `
 x ) ) )  ->  -.  A. x  e.  dom  A ( A `
 x )  =  ( B `  x
) )
17 df-ne 2608 . . . . . . . . . . . . 13  |-  ( ( A `  x )  =/=  ( B `  x )  <->  -.  ( A `  x )  =  ( B `  x ) )
1817rexbii 2740 . . . . . . . . . . . 12  |-  ( E. x  e.  dom  A
( A `  x
)  =/=  ( B `
 x )  <->  E. x  e.  dom  A  -.  ( A `  x )  =  ( B `  x ) )
19 rexnal 2726 . . . . . . . . . . . 12  |-  ( E. x  e.  dom  A  -.  ( A `  x
)  =  ( B `
 x )  <->  -.  A. x  e.  dom  A ( A `
 x )  =  ( B `  x
) )
2018, 19bitri 249 . . . . . . . . . . 11  |-  ( E. x  e.  dom  A
( A `  x
)  =/=  ( B `
 x )  <->  -.  A. x  e.  dom  A ( A `
 x )  =  ( B `  x
) )
21 nodenselem4 27825 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  A <s
B )  ->  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  On )
22 eloni 4729 . . . . . . . . . . . . . . 15  |-  ( |^| { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) }  e.  On  ->  Ord  |^|
{ a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )
2321, 22syl 16 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  A <s
B )  ->  Ord  |^|
{ a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )
2423adantr 465 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  No  /\  B  e.  No )  /\  A <s B )  /\  ( x  e.  dom  A  /\  ( A `  x )  =/=  ( B `  x )
) )  ->  Ord  |^|
{ a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )
25 nodmord 27794 . . . . . . . . . . . . . 14  |-  ( A  e.  No  ->  Ord  dom 
A )
2625ad3antrrr 729 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  No  /\  B  e.  No )  /\  A <s B )  /\  ( x  e.  dom  A  /\  ( A `  x )  =/=  ( B `  x )
) )  ->  Ord  dom 
A )
27 nodmon 27791 . . . . . . . . . . . . . . . . . . 19  |-  ( A  e.  No  ->  dom  A  e.  On )
28 onelon 4744 . . . . . . . . . . . . . . . . . . 19  |-  ( ( dom  A  e.  On  /\  x  e.  dom  A
)  ->  x  e.  On )
2927, 28sylan 471 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  e.  No  /\  x  e.  dom  A )  ->  x  e.  On )
3029ex 434 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  No  ->  (
x  e.  dom  A  ->  x  e.  On ) )
3130ad2antrr 725 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  A <s
B )  ->  (
x  e.  dom  A  ->  x  e.  On ) )
3231anim1d 564 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  A <s
B )  ->  (
( x  e.  dom  A  /\  ( A `  x )  =/=  ( B `  x )
)  ->  ( x  e.  On  /\  ( A `
 x )  =/=  ( B `  x
) ) ) )
3332imp 429 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  No  /\  B  e.  No )  /\  A <s B )  /\  ( x  e.  dom  A  /\  ( A `  x )  =/=  ( B `  x )
) )  ->  (
x  e.  On  /\  ( A `  x )  =/=  ( B `  x ) ) )
34 fveq2 5691 . . . . . . . . . . . . . . . 16  |-  ( a  =  x  ->  ( A `  a )  =  ( A `  x ) )
35 fveq2 5691 . . . . . . . . . . . . . . . 16  |-  ( a  =  x  ->  ( B `  a )  =  ( B `  x ) )
3634, 35neeq12d 2623 . . . . . . . . . . . . . . 15  |-  ( a  =  x  ->  (
( A `  a
)  =/=  ( B `
 a )  <->  ( A `  x )  =/=  ( B `  x )
) )
3736intminss 4154 . . . . . . . . . . . . . 14  |-  ( ( x  e.  On  /\  ( A `  x )  =/=  ( B `  x ) )  ->  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  C_  x )
3833, 37syl 16 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  No  /\  B  e.  No )  /\  A <s B )  /\  ( x  e.  dom  A  /\  ( A `  x )  =/=  ( B `  x )
) )  ->  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  C_  x )
39 simprl 755 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  No  /\  B  e.  No )  /\  A <s B )  /\  ( x  e.  dom  A  /\  ( A `  x )  =/=  ( B `  x )
) )  ->  x  e.  dom  A )
40 ordtr2 4763 . . . . . . . . . . . . . 14  |-  ( ( Ord  |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) }  /\  Ord  dom 
A )  ->  (
( |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) }  C_  x  /\  x  e.  dom  A )  ->  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  dom  A ) )
4140imp 429 . . . . . . . . . . . . 13  |-  ( ( ( Ord  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  /\  Ord  dom  A )  /\  ( |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) }  C_  x  /\  x  e.  dom  A ) )  ->  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  dom  A )
4224, 26, 38, 39, 41syl22anc 1219 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  No  /\  B  e.  No )  /\  A <s B )  /\  ( x  e.  dom  A  /\  ( A `  x )  =/=  ( B `  x )
) )  ->  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  dom  A )
4342rexlimdvaa 2842 . . . . . . . . . . 11  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  A <s
B )  ->  ( E. x  e.  dom  A ( A `  x
)  =/=  ( B `
 x )  ->  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  dom  A ) )
4420, 43syl5bir 218 . . . . . . . . . 10  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  A <s
B )  ->  ( -.  A. x  e.  dom  A ( A `  x
)  =  ( B `
 x )  ->  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  dom  A ) )
4516, 44syl5 32 . . . . . . . . 9  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  A <s
B )  ->  (
( dom  A  =  dom  B  /\  -.  ( dom  A  =  dom  B  /\  A. x  e.  dom  A ( A `  x
)  =  ( B `
 x ) ) )  ->  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  dom  A ) )
4645exp4b 607 . . . . . . . 8  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( A <s
B  ->  ( dom  A  =  dom  B  -> 
( -.  ( dom 
A  =  dom  B  /\  A. x  e.  dom  A ( A `  x
)  =  ( B `
 x ) )  ->  |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) }  e.  dom  A ) ) ) )
4746com23 78 . . . . . . 7  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( dom  A  =  dom  B  ->  ( A <s B  -> 
( -.  ( dom 
A  =  dom  B  /\  A. x  e.  dom  A ( A `  x
)  =  ( B `
 x ) )  ->  |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) }  e.  dom  A ) ) ) )
4847imp32 433 . . . . . 6  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( dom  A  =  dom  B  /\  A <s B ) )  ->  ( -.  ( dom  A  =  dom  B  /\  A. x  e.  dom  A ( A `  x
)  =  ( B `
 x ) )  ->  |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) }  e.  dom  A ) )
4913, 48sylbid 215 . . . . 5  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( dom  A  =  dom  B  /\  A <s B ) )  ->  ( -.  A  =  B  ->  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  dom  A ) )
507, 49mpd 15 . . . 4  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( dom  A  =  dom  B  /\  A <s B ) )  ->  |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) }  e.  dom  A )
5150ex 434 . . 3  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( ( dom  A  =  dom  B  /\  A <s B )  ->  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  dom  A ) )
52 bdayval 27789 . . . . 5  |-  ( A  e.  No  ->  ( bday `  A )  =  dom  A )
53 bdayval 27789 . . . . 5  |-  ( B  e.  No  ->  ( bday `  B )  =  dom  B )
5452, 53eqeqan12d 2458 . . . 4  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( ( bday `  A
)  =  ( bday `  B )  <->  dom  A  =  dom  B ) )
5554anbi1d 704 . . 3  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( ( ( bday `  A )  =  (
bday `  B )  /\  A <s B )  <->  ( dom  A  =  dom  B  /\  A <s B ) ) )
5652eleq2d 2510 . . . 4  |-  ( A  e.  No  ->  ( |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  ( bday `  A )  <->  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  dom  A ) )
5756adantr 465 . . 3  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  ( bday `  A )  <->  |^|
{ a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  dom  A ) )
5851, 55, 573imtr4d 268 . 2  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( ( ( bday `  A )  =  (
bday `  B )  /\  A <s B )  ->  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  ( bday `  A )
) )
5958imp 429 1  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( ( bday `  A )  =  (
bday `  B )  /\  A <s B ) )  ->  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  ( bday `  A )
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2606   A.wral 2715   E.wrex 2716   {crab 2719    C_ wss 3328   |^|cint 4128   class class class wbr 4292   Ord word 4718   Oncon0 4719   dom cdm 4840   Fun wfun 5412   ` cfv 5418   Nocsur 27781   <scslt 27782   bdaycbday 27783
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-1o 6920  df-2o 6921  df-no 27784  df-slt 27785  df-bday 27786
This theorem is referenced by:  nodenselem6  27827  nodenselem8  27829  nodense  27830
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