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Theorem nodenselem5 30574
Description: Lemma for nodense 30578. If the birthdays of two distinct surreals are equal, then the ordinal from nodenselem4 30573 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 30559 . . . . . . . . 9  |-  ( A  e.  No  ->  -.  A <s A )
2 breq2 4406 . . . . . . . . . 10  |-  ( A  =  B  ->  ( A <s A  <->  A <s B ) )
32notbid 296 . . . . . . . . 9  |-  ( A  =  B  ->  ( -.  A <s A  <->  -.  A <s B ) )
41, 3syl5ibcom 224 . . . . . . . 8  |-  ( A  e.  No  ->  ( A  =  B  ->  -.  A <s B ) )
54con2d 119 . . . . . . 7  |-  ( A  e.  No  ->  ( A <s B  ->  -.  A  =  B
) )
65imp 431 . . . . . 6  |-  ( ( A  e.  No  /\  A <s B )  ->  -.  A  =  B )
76ad2ant2rl 755 . . . . 5  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( dom  A  =  dom  B  /\  A <s B ) )  ->  -.  A  =  B )
8 nofun 30536 . . . . . . . . 9  |-  ( A  e.  No  ->  Fun  A )
9 nofun 30536 . . . . . . . . 9  |-  ( B  e.  No  ->  Fun  B )
10 eqfunfv 5981 . . . . . . . . 9  |-  ( ( Fun  A  /\  Fun  B )  ->  ( A  =  B  <->  ( dom  A  =  dom  B  /\  A. x  e.  dom  A ( A `  x )  =  ( B `  x ) ) ) )
118, 9, 10syl2an 480 . . . . . . . 8  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( A  =  B  <-> 
( dom  A  =  dom  B  /\  A. x  e.  dom  A ( A `
 x )  =  ( B `  x
) ) ) )
1211notbid 296 . . . . . . 7  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( -.  A  =  B  <->  -.  ( dom  A  =  dom  B  /\  A. x  e.  dom  A
( A `  x
)  =  ( B `
 x ) ) ) )
1312adantr 467 . . . . . 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 424 . . . . . . . . . . . 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 210 . . . . . . . . . . 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 432 . . . . . . . . . 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 2624 . . . . . . . . . . . . 13  |-  ( ( A `  x )  =/=  ( B `  x )  <->  -.  ( A `  x )  =  ( B `  x ) )
1817rexbii 2889 . . . . . . . . . . . 12  |-  ( E. x  e.  dom  A
( A `  x
)  =/=  ( B `
 x )  <->  E. x  e.  dom  A  -.  ( A `  x )  =  ( B `  x ) )
19 rexnal 2836 . . . . . . . . . . . 12  |-  ( E. x  e.  dom  A  -.  ( A `  x
)  =  ( B `
 x )  <->  -.  A. x  e.  dom  A ( A `
 x )  =  ( B `  x
) )
2018, 19bitri 253 . . . . . . . . . . 11  |-  ( E. x  e.  dom  A
( A `  x
)  =/=  ( B `
 x )  <->  -.  A. x  e.  dom  A ( A `
 x )  =  ( B `  x
) )
21 nodenselem4 30573 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  A <s
B )  ->  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  On )
22 eloni 5433 . . . . . . . . . . . . . . 15  |-  ( |^| { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) }  e.  On  ->  Ord  |^|
{ a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )
2321, 22syl 17 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  A <s
B )  ->  Ord  |^|
{ a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )
2423adantr 467 . . . . . . . . . . . . 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 30540 . . . . . . . . . . . . . 14  |-  ( A  e.  No  ->  Ord  dom 
A )
2625ad3antrrr 736 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  No  /\  B  e.  No )  /\  A <s B )  /\  ( x  e.  dom  A  /\  ( A `  x )  =/=  ( B `  x )
) )  ->  Ord  dom 
A )
27 nodmon 30537 . . . . . . . . . . . . . . . . . . 19  |-  ( A  e.  No  ->  dom  A  e.  On )
28 onelon 5448 . . . . . . . . . . . . . . . . . . 19  |-  ( ( dom  A  e.  On  /\  x  e.  dom  A
)  ->  x  e.  On )
2927, 28sylan 474 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  e.  No  /\  x  e.  dom  A )  ->  x  e.  On )
3029ex 436 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  No  ->  (
x  e.  dom  A  ->  x  e.  On ) )
3130ad2antrr 732 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  A <s
B )  ->  (
x  e.  dom  A  ->  x  e.  On ) )
3231anim1d 568 . . . . . . . . . . . . . . 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 431 . . . . . . . . . . . . . 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 5865 . . . . . . . . . . . . . . . 16  |-  ( a  =  x  ->  ( A `  a )  =  ( A `  x ) )
35 fveq2 5865 . . . . . . . . . . . . . . . 16  |-  ( a  =  x  ->  ( B `  a )  =  ( B `  x ) )
3634, 35neeq12d 2685 . . . . . . . . . . . . . . 15  |-  ( a  =  x  ->  (
( A `  a
)  =/=  ( B `
 a )  <->  ( A `  x )  =/=  ( B `  x )
) )
3736intminss 4261 . . . . . . . . . . . . . 14  |-  ( ( x  e.  On  /\  ( A `  x )  =/=  ( B `  x ) )  ->  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  C_  x )
3833, 37syl 17 . . . . . . . . . . . . 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 764 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  No  /\  B  e.  No )  /\  A <s B )  /\  ( x  e.  dom  A  /\  ( A `  x )  =/=  ( B `  x )
) )  ->  x  e.  dom  A )
40 ordtr2 5467 . . . . . . . . . . . . . 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 431 . . . . . . . . . . . . 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 1269 . . . . . . . . . . . 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 2880 . . . . . . . . . . 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 222 . . . . . . . . . 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 33 . . . . . . . . 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 612 . . . . . . . 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 81 . . . . . . 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 435 . . . . . 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 219 . . . . 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 436 . . 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 30535 . . . . 5  |-  ( A  e.  No  ->  ( bday `  A )  =  dom  A )
53 bdayval 30535 . . . . 5  |-  ( B  e.  No  ->  ( bday `  B )  =  dom  B )
5452, 53eqeqan12d 2467 . . . 4  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( ( bday `  A
)  =  ( bday `  B )  <->  dom  A  =  dom  B ) )
5554anbi1d 711 . . 3  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( ( ( bday `  A )  =  (
bday `  B )  /\  A <s B )  <->  ( dom  A  =  dom  B  /\  A <s B ) ) )
5652eleq2d 2514 . . . 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 467 . . 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 272 . 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 431 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 188    /\ wa 371    = wceq 1444    e. wcel 1887    =/= wne 2622   A.wral 2737   E.wrex 2738   {crab 2741    C_ wss 3404   |^|cint 4234   class class class wbr 4402   dom cdm 4834   Ord word 5422   Oncon0 5423   Fun wfun 5576   ` cfv 5582   Nocsur 30527   <scslt 30528   bdaycbday 30529
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-ord 5426  df-on 5427  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-1o 7182  df-2o 7183  df-no 30530  df-slt 30531  df-bday 30532
This theorem is referenced by:  nodenselem6  30575  nodenselem8  30577  nodense  30578
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