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Theorem nodenselem7 30160
Description: Lemma for nodense 30162. 
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 30157 . . . . 5  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  A <s
B )  ->  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  On )
21adantrl 716 . . . 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 5437 . . . . 5  |-  ( (
|^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  On  /\  C  e.  |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } )  ->  C  e.  On )
43ex 434 . . . 4  |-  ( |^| { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) }  e.  On  ->  ( C  e.  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  ->  C  e.  On ) )
52, 4syl 17 . . 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 471 . . . . . . . 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 5446 . . . . . . . . . . . . . 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 329 . . . . . . . . . . . . 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 209 . . . . . . . . . . . 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 434 . . . . . . . . . . 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 80 . . . . . . . . . 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 17 . . . . . . . . 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 429 . . . . . . . 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 4244 . . . . . . 7  |-  ( C  e.  { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) }  ->  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  C_  C )
1614, 15nsyl 123 . . . . . 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 434 . . . . 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 5851 . . . . . . . 8  |-  ( a  =  C  ->  ( A `  a )  =  ( A `  C ) )
19 fveq2 5851 . . . . . . . 8  |-  ( a  =  C  ->  ( B `  a )  =  ( B `  C ) )
2018, 19neeq12d 2684 . . . . . . 7  |-  ( a  =  C  ->  (
( A `  a
)  =/=  ( B `
 a )  <->  ( A `  C )  =/=  ( B `  C )
) )
2120elrab 3209 . . . . . 6  |-  ( C  e.  { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) }  <->  ( C  e.  On  /\  ( A `
 C )  =/=  ( B `  C
) ) )
2221notbii 296 . . . . 5  |-  ( -.  C  e.  { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  <->  -.  ( C  e.  On  /\  ( A `  C )  =/=  ( B `  C
) ) )
2317, 22syl6ib 228 . . . 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 422 . . . 4  |-  ( ( C  e.  On  ->  -.  ( A `  C
)  =/=  ( B `
 C ) )  <->  -.  ( C  e.  On  /\  ( A `  C
)  =/=  ( B `
 C ) ) )
2523, 24syl6ibr 229 . . 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 40 . 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 2602 . . 3  |-  ( ( A `  C )  =/=  ( B `  C )  <->  -.  ( A `  C )  =  ( B `  C ) )
2827con2bii 332 . 2  |-  ( ( A `  C )  =  ( B `  C )  <->  -.  ( A `  C )  =/=  ( B `  C
) )
2926, 28syl6ibr 229 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1407    e. wcel 1844    =/= wne 2600   {crab 2760    C_ wss 3416   |^|cint 4229   class class class wbr 4397   Oncon0 5412   ` cfv 5571   Nocsur 30113   <scslt 30114   bdaycbday 30115
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-tp 3979  df-op 3981  df-uni 4194  df-int 4230  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-tr 4492  df-eprel 4736  df-id 4740  df-po 4746  df-so 4747  df-fr 4784  df-we 4786  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-ord 5415  df-on 5416  df-suc 5418  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-1o 7169  df-2o 7170  df-no 30116  df-slt 30117  df-bday 30118
This theorem is referenced by:  nodense  30162
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