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Theorem nodenselem7 29024
Description: Lemma for nodense 29026. 
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 29021 . . . . 5  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  A <s
B )  ->  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  On )
21adantrl 715 . . . 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 4903 . . . . 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 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 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 4912 . . . . . . . . . . . . . 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 207 . . . . . . . . . . . 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 78 . . . . . . . . . 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 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 4297 . . . . . . 7  |-  ( C  e.  { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) }  ->  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  C_  C )
1614, 15nsyl 121 . . . . . 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 5864 . . . . . . . 8  |-  ( a  =  C  ->  ( A `  a )  =  ( A `  C ) )
19 fveq2 5864 . . . . . . . 8  |-  ( a  =  C  ->  ( B `  a )  =  ( B `  C ) )
2018, 19neeq12d 2746 . . . . . . 7  |-  ( a  =  C  ->  (
( A `  a
)  =/=  ( B `
 a )  <->  ( A `  C )  =/=  ( B `  C )
) )
2120elrab 3261 . . . . . 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 226 . . . 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 227 . . 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 2664 . . 3  |-  ( ( A `  C )  =/=  ( B `  C )  <->  -.  ( A `  C )  =  ( B `  C ) )
2827con2bii 332 . 2  |-  ( ( A `  C )  =  ( B `  C )  <->  -.  ( A `  C )  =/=  ( B `  C
) )
2926, 28syl6ibr 227 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 1379    e. wcel 1767    =/= wne 2662   {crab 2818    C_ wss 3476   |^|cint 4282   class class class wbr 4447   Oncon0 4878   ` cfv 5586   Nocsur 28977   <scslt 28978   bdaycbday 28979
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-1o 7127  df-2o 7128  df-no 28980  df-slt 28981  df-bday 28982
This theorem is referenced by:  nodense  29026
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