Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  nodenselem7 Structured version   Unicode version

Theorem nodenselem7 25642
Description: Lemma for nodense 25644. 
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 25639 . . . . 5  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  A < s B )  ->  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  On )
21adantrl 697 . . . 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 4606 . . . . 5  |-  ( (
|^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  On  /\  C  e.  |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } )  ->  C  e.  On )
43ex 424 . . . 4  |-  ( |^| { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) }  e.  On  ->  ( C  e.  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  ->  C  e.  On ) )
52, 4syl 16 . . 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 458 . . . . . . . 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 4615 . . . . . . . . . . . . . 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 320 . . . . . . . . . . . . 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 199 . . . . . . . . . . . 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 424 . . . . . . . . . . 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 74 . . . . . . . . . 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 16 . . . . . . . . 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 419 . . . . . . . 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 15 . . . . . . 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 4065 . . . . . . 7  |-  ( C  e.  { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) }  ->  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  C_  C )
1614, 15nsyl 115 . . . . . 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 424 . . . . 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 5728 . . . . . . . 8  |-  ( a  =  C  ->  ( A `  a )  =  ( A `  C ) )
19 fveq2 5728 . . . . . . . 8  |-  ( a  =  C  ->  ( B `  a )  =  ( B `  C ) )
2018, 19neeq12d 2616 . . . . . . 7  |-  ( a  =  C  ->  (
( A `  a
)  =/=  ( B `
 a )  <->  ( A `  C )  =/=  ( B `  C )
) )
2120elrab 3092 . . . . . 6  |-  ( C  e.  { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) }  <->  ( C  e.  On  /\  ( A `
 C )  =/=  ( B `  C
) ) )
2221notbii 288 . . . . 5  |-  ( -.  C  e.  { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  <->  -.  ( C  e.  On  /\  ( A `  C )  =/=  ( B `  C
) ) )
2317, 22syl6ib 218 . . . 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 412 . . . 4  |-  ( ( C  e.  On  ->  -.  ( A `  C
)  =/=  ( B `
 C ) )  <->  -.  ( C  e.  On  /\  ( A `  C
)  =/=  ( B `
 C ) ) )
2523, 24syl6ibr 219 . . 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 38 . 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 2601 . . 3  |-  ( ( A `  C )  =/=  ( B `  C )  <->  -.  ( A `  C )  =  ( B `  C ) )
2827con2bii 323 . 2  |-  ( ( A `  C )  =  ( B `  C )  <->  -.  ( A `  C )  =/=  ( B `  C
) )
2926, 28syl6ibr 219 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 359    = wceq 1652    e. wcel 1725    =/= wne 2599   {crab 2709    C_ wss 3320   |^|cint 4050   class class class wbr 4212   Oncon0 4581   ` cfv 5454   Nocsur 25595   < scslt 25596   bdaycbday 25597
This theorem is referenced by:  nodense  25644
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-suc 4587  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-1o 6724  df-2o 6725  df-no 25598  df-slt 25599  df-bday 25600
  Copyright terms: Public domain W3C validator