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Theorem nodenselem7 24343
Description: Lemma for nodense 24345. 
A and  B are equal at all elements of the abstraction. (Contributed by Scott Fenton, 17-Jun-2011.)
Assertion
Ref Expression
nodenselem7  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( ( bday `  A )  =  (
bday `  B )  /\  A < s B ) )  ->  ( C  e.  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  ->  ( A `  C )  =  ( B `  C ) ) )
Distinct variable groups:    A, a    B, a    C, a

Proof of Theorem nodenselem7
StepHypRef Expression
1 nodenselem4 24340 . . . . 5  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  A < s B )  ->  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  On )
21adantrl 696 . . . 4  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( ( bday `  A )  =  (
bday `  B )  /\  A < s B ) )  ->  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  On )
3 onelon 4419 . . . . 5  |-  ( (
|^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  On  /\  C  e.  |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } )  ->  C  e.  On )
43ex 423 . . . 4  |-  ( |^| { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) }  e.  On  ->  ( C  e.  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  ->  C  e.  On ) )
52, 4syl 15 . . 3  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( ( bday `  A )  =  (
bday `  B )  /\  A < s B ) )  ->  ( C  e.  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  ->  C  e.  On ) )
62, 3sylan 457 . . . . . . . 8  |-  ( ( ( ( A  e.  No  /\  B  e.  No )  /\  (
( bday `  A )  =  ( bday `  B
)  /\  A < s B ) )  /\  C  e.  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  ->  C  e.  On )
7 ontri1 4428 . . . . . . . . . . . . . 14  |-  ( (
|^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  On  /\  C  e.  On )  ->  ( |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  C_  C  <->  -.  C  e.  |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } ) )
87con2bid 319 . . . . . . . . . . . . 13  |-  ( (
|^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  On  /\  C  e.  On )  ->  ( C  e.  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  <->  -.  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  C_  C ) )
98biimpd 198 . . . . . . . . . . . 12  |-  ( (
|^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  On  /\  C  e.  On )  ->  ( C  e.  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  ->  -. 
|^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  C_  C ) )
109ex 423 . . . . . . . . . . 11  |-  ( |^| { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) }  e.  On  ->  ( C  e.  On  ->  ( C  e.  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  ->  -. 
|^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  C_  C ) ) )
1110com23 72 . . . . . . . . . 10  |-  ( |^| { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) }  e.  On  ->  ( C  e.  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  ->  ( C  e.  On  ->  -. 
|^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  C_  C ) ) )
122, 11syl 15 . . . . . . . . 9  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( ( bday `  A )  =  (
bday `  B )  /\  A < s B ) )  ->  ( C  e.  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  ->  ( C  e.  On  ->  -. 
|^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  C_  C ) ) )
1312imp 418 . . . . . . . 8  |-  ( ( ( ( A  e.  No  /\  B  e.  No )  /\  (
( bday `  A )  =  ( bday `  B
)  /\  A < s B ) )  /\  C  e.  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  ->  ( C  e.  On  ->  -.  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  C_  C ) )
146, 13mpd 14 . . . . . . 7  |-  ( ( ( ( A  e.  No  /\  B  e.  No )  /\  (
( bday `  A )  =  ( bday `  B
)  /\  A < s B ) )  /\  C  e.  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  ->  -.  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  C_  C )
15 intss1 3879 . . . . . . 7  |-  ( C  e.  { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) }  ->  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  C_  C )
1614, 15nsyl 113 . . . . . 6  |-  ( ( ( ( A  e.  No  /\  B  e.  No )  /\  (
( bday `  A )  =  ( bday `  B
)  /\  A < s B ) )  /\  C  e.  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  ->  -.  C  e.  { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) } )
1716ex 423 . . . . 5  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( ( bday `  A )  =  (
bday `  B )  /\  A < s B ) )  ->  ( C  e.  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  ->  -.  C  e.  { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) )
18 fveq2 5527 . . . . . . . 8  |-  ( a  =  C  ->  ( A `  a )  =  ( A `  C ) )
19 fveq2 5527 . . . . . . . 8  |-  ( a  =  C  ->  ( B `  a )  =  ( B `  C ) )
2018, 19neeq12d 2463 . . . . . . 7  |-  ( a  =  C  ->  (
( A `  a
)  =/=  ( B `
 a )  <->  ( A `  C )  =/=  ( B `  C )
) )
2120elrab 2925 . . . . . 6  |-  ( C  e.  { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) }  <->  ( C  e.  On  /\  ( A `
 C )  =/=  ( B `  C
) ) )
2221notbii 287 . . . . 5  |-  ( -.  C  e.  { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  <->  -.  ( C  e.  On  /\  ( A `  C )  =/=  ( B `  C
) ) )
2317, 22syl6ib 217 . . . 4  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( ( bday `  A )  =  (
bday `  B )  /\  A < s B ) )  ->  ( C  e.  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  ->  -.  ( C  e.  On  /\  ( A `  C
)  =/=  ( B `
 C ) ) ) )
24 imnan 411 . . . 4  |-  ( ( C  e.  On  ->  -.  ( A `  C
)  =/=  ( B `
 C ) )  <->  -.  ( C  e.  On  /\  ( A `  C
)  =/=  ( B `
 C ) ) )
2523, 24syl6ibr 218 . . 3  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( ( bday `  A )  =  (
bday `  B )  /\  A < s B ) )  ->  ( C  e.  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  ->  ( C  e.  On  ->  -.  ( A `  C
)  =/=  ( B `
 C ) ) ) )
265, 25mpdd 36 . 2  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( ( bday `  A )  =  (
bday `  B )  /\  A < s B ) )  ->  ( C  e.  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  ->  -.  ( A `  C
)  =/=  ( B `
 C ) ) )
27 df-ne 2450 . . 3  |-  ( ( A `  C )  =/=  ( B `  C )  <->  -.  ( A `  C )  =  ( B `  C ) )
2827con2bii 322 . 2  |-  ( ( A `  C )  =  ( B `  C )  <->  -.  ( A `  C )  =/=  ( B `  C
) )
2926, 28syl6ibr 218 1  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( ( bday `  A )  =  (
bday `  B )  /\  A < s B ) )  ->  ( C  e.  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  ->  ( A `  C )  =  ( B `  C ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1625    e. wcel 1686    =/= wne 2448   {crab 2549    C_ wss 3154   |^|cint 3864   class class class wbr 4025   Oncon0 4394   ` cfv 5257   Nocsur 24296   < scslt 24297   bdaycbday 24298
This theorem is referenced by:  nodense  24345
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-reu 2552  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-int 3865  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-we 4356  df-ord 4397  df-on 4398  df-suc 4400  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-1o 6481  df-2o 6482  df-no 24299  df-slt 24300  df-bday 24301
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