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Theorem nodenselem7 25082
Description: Lemma for nodense 25084. 
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 25079 . . . . 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 4520 . . . . 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 4529 . . . . . . . . . . . . . 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 3979 . . . . . . 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 5632 . . . . . . . 8  |-  ( a  =  C  ->  ( A `  a )  =  ( A `  C ) )
19 fveq2 5632 . . . . . . . 8  |-  ( a  =  C  ->  ( B `  a )  =  ( B `  C ) )
2018, 19neeq12d 2544 . . . . . . 7  |-  ( a  =  C  ->  (
( A `  a
)  =/=  ( B `
 a )  <->  ( A `  C )  =/=  ( B `  C )
) )
2120elrab 3009 . . . . . 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 2531 . . 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 1647    e. wcel 1715    =/= wne 2529   {crab 2632    C_ wss 3238   |^|cint 3964   class class class wbr 4125   Oncon0 4495   ` cfv 5358   Nocsur 25035   < scslt 25036   bdaycbday 25037
This theorem is referenced by:  nodense  25084
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-reu 2635  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-int 3965  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-we 4457  df-ord 4498  df-on 4499  df-suc 4501  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-1o 6621  df-2o 6622  df-no 25038  df-slt 25039  df-bday 25040
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