Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  nobndlem6 Unicode version

Theorem nobndlem6 24353
Description: Lemma for nobndup 24356 and nobnddown 24357. Given an element  A of  F, then the first position where it differs from  X is strictly less than  C (Contributed by Scott Fenton, 3-Aug-2011.)
Hypotheses
Ref Expression
nobndlem6.1  |-  X  e. 
{ 1o ,  2o }
nobndlem6.2  |-  C  = 
|^| { a  e.  On  |  A. n  e.  F  E. b  e.  a 
( n `  b
)  =/=  X }
Assertion
Ref Expression
nobndlem6  |-  ( ( F  C_  No  /\  A  e.  F )  ->  |^| { x  e.  On  |  ( A `
 x )  =/= 
X }  e.  C
)
Distinct variable groups:    A, a,
b, x    F, a,
b    X, a, b, x   
n, X, a, b    A, n    n, F
Allowed substitution hints:    C( x, n, a, b)    F( x)

Proof of Theorem nobndlem6
StepHypRef Expression
1 bdayelon 24336 . . . . 5  |-  ( bday `  A )  e.  On
2 ssel2 3177 . . . . . 6  |-  ( ( F  C_  No  /\  A  e.  F )  ->  A  e.  No )
3 nobndlem6.1 . . . . . . . 8  |-  X  e. 
{ 1o ,  2o }
43nosgnn0i 24315 . . . . . . 7  |-  (/)  =/=  X
5 fvnobday 24338 . . . . . . . 8  |-  ( A  e.  No  ->  ( A `  ( bday `  A ) )  =  (/) )
65neeq1d 2461 . . . . . . 7  |-  ( A  e.  No  ->  (
( A `  ( bday `  A ) )  =/=  X  <->  (/)  =/=  X
) )
74, 6mpbiri 224 . . . . . 6  |-  ( A  e.  No  ->  ( A `  ( bday `  A ) )  =/= 
X )
82, 7syl 15 . . . . 5  |-  ( ( F  C_  No  /\  A  e.  F )  ->  ( A `  ( bday `  A ) )  =/= 
X )
9 fveq2 5527 . . . . . . 7  |-  ( x  =  ( bday `  A
)  ->  ( A `  x )  =  ( A `  ( bday `  A ) ) )
109neeq1d 2461 . . . . . 6  |-  ( x  =  ( bday `  A
)  ->  ( ( A `  x )  =/=  X  <->  ( A `  ( bday `  A )
)  =/=  X ) )
1110rspcev 2886 . . . . 5  |-  ( ( ( bday `  A
)  e.  On  /\  ( A `  ( bday `  A ) )  =/= 
X )  ->  E. x  e.  On  ( A `  x )  =/=  X
)
121, 8, 11sylancr 644 . . . 4  |-  ( ( F  C_  No  /\  A  e.  F )  ->  E. x  e.  On  ( A `  x )  =/=  X
)
13 onintrab2 4595 . . . 4  |-  ( E. x  e.  On  ( A `  x )  =/=  X  <->  |^| { x  e.  On  |  ( A `
 x )  =/= 
X }  e.  On )
1412, 13sylib 188 . . 3  |-  ( ( F  C_  No  /\  A  e.  F )  ->  |^| { x  e.  On  |  ( A `
 x )  =/= 
X }  e.  On )
15 fveq1 5526 . . . . . . . . 9  |-  ( n  =  A  ->  (
n `  b )  =  ( A `  b ) )
1615neeq1d 2461 . . . . . . . 8  |-  ( n  =  A  ->  (
( n `  b
)  =/=  X  <->  ( A `  b )  =/=  X
) )
1716rexbidv 2566 . . . . . . 7  |-  ( n  =  A  ->  ( E. b  e.  a 
( n `  b
)  =/=  X  <->  E. b  e.  a  ( A `  b )  =/=  X
) )
1817rspcv 2882 . . . . . 6  |-  ( A  e.  F  ->  ( A. n  e.  F  E. b  e.  a 
( n `  b
)  =/=  X  ->  E. b  e.  a 
( A `  b
)  =/=  X ) )
1918ad2antlr 707 . . . . 5  |-  ( ( ( F  C_  No  /\  A  e.  F )  /\  a  e.  On )  ->  ( A. n  e.  F  E. b  e.  a  ( n `  b )  =/=  X  ->  E. b  e.  a  ( A `  b
)  =/=  X ) )
2014ad2antrr 706 . . . . . . . 8  |-  ( ( ( ( F  C_  No  /\  A  e.  F
)  /\  a  e.  On )  /\  (
b  e.  a  /\  ( A `  b )  =/=  X ) )  ->  |^| { x  e.  On  |  ( A `
 x )  =/= 
X }  e.  On )
21 simplr 731 . . . . . . . 8  |-  ( ( ( ( F  C_  No  /\  A  e.  F
)  /\  a  e.  On )  /\  (
b  e.  a  /\  ( A `  b )  =/=  X ) )  ->  a  e.  On )
22 onelon 4419 . . . . . . . . . . . 12  |-  ( ( a  e.  On  /\  b  e.  a )  ->  b  e.  On )
2322anim1i 551 . . . . . . . . . . 11  |-  ( ( ( a  e.  On  /\  b  e.  a )  /\  ( A `  b )  =/=  X
)  ->  ( b  e.  On  /\  ( A `
 b )  =/= 
X ) )
2423anasss 628 . . . . . . . . . 10  |-  ( ( a  e.  On  /\  ( b  e.  a  /\  ( A `  b )  =/=  X
) )  ->  (
b  e.  On  /\  ( A `  b )  =/=  X ) )
25 fveq2 5527 . . . . . . . . . . . 12  |-  ( x  =  b  ->  ( A `  x )  =  ( A `  b ) )
2625neeq1d 2461 . . . . . . . . . . 11  |-  ( x  =  b  ->  (
( A `  x
)  =/=  X  <->  ( A `  b )  =/=  X
) )
2726intminss 3890 . . . . . . . . . 10  |-  ( ( b  e.  On  /\  ( A `  b )  =/=  X )  ->  |^| { x  e.  On  |  ( A `  x )  =/=  X }  C_  b )
2824, 27syl 15 . . . . . . . . 9  |-  ( ( a  e.  On  /\  ( b  e.  a  /\  ( A `  b )  =/=  X
) )  ->  |^| { x  e.  On  |  ( A `
 x )  =/= 
X }  C_  b
)
2928adantll 694 . . . . . . . 8  |-  ( ( ( ( F  C_  No  /\  A  e.  F
)  /\  a  e.  On )  /\  (
b  e.  a  /\  ( A `  b )  =/=  X ) )  ->  |^| { x  e.  On  |  ( A `
 x )  =/= 
X }  C_  b
)
30 simprl 732 . . . . . . . 8  |-  ( ( ( ( F  C_  No  /\  A  e.  F
)  /\  a  e.  On )  /\  (
b  e.  a  /\  ( A `  b )  =/=  X ) )  ->  b  e.  a )
31 ontr2 4441 . . . . . . . . 9  |-  ( (
|^| { x  e.  On  |  ( A `  x )  =/=  X }  e.  On  /\  a  e.  On )  ->  (
( |^| { x  e.  On  |  ( A `
 x )  =/= 
X }  C_  b  /\  b  e.  a
)  ->  |^| { x  e.  On  |  ( A `
 x )  =/= 
X }  e.  a ) )
3231imp 418 . . . . . . . 8  |-  ( ( ( |^| { x  e.  On  |  ( A `
 x )  =/= 
X }  e.  On  /\  a  e.  On )  /\  ( |^| { x  e.  On  |  ( A `
 x )  =/= 
X }  C_  b  /\  b  e.  a
) )  ->  |^| { x  e.  On  |  ( A `
 x )  =/= 
X }  e.  a )
3320, 21, 29, 30, 32syl22anc 1183 . . . . . . 7  |-  ( ( ( ( F  C_  No  /\  A  e.  F
)  /\  a  e.  On )  /\  (
b  e.  a  /\  ( A `  b )  =/=  X ) )  ->  |^| { x  e.  On  |  ( A `
 x )  =/= 
X }  e.  a )
3433exp32 588 . . . . . 6  |-  ( ( ( F  C_  No  /\  A  e.  F )  /\  a  e.  On )  ->  ( b  e.  a  ->  ( ( A `  b )  =/=  X  ->  |^| { x  e.  On  |  ( A `
 x )  =/= 
X }  e.  a ) ) )
3534rexlimdv 2668 . . . . 5  |-  ( ( ( F  C_  No  /\  A  e.  F )  /\  a  e.  On )  ->  ( E. b  e.  a  ( A `  b )  =/=  X  ->  |^| { x  e.  On  |  ( A `
 x )  =/= 
X }  e.  a ) )
3619, 35syld 40 . . . 4  |-  ( ( ( F  C_  No  /\  A  e.  F )  /\  a  e.  On )  ->  ( A. n  e.  F  E. b  e.  a  ( n `  b )  =/=  X  ->  |^| { x  e.  On  |  ( A `
 x )  =/= 
X }  e.  a ) )
3736ralrimiva 2628 . . 3  |-  ( ( F  C_  No  /\  A  e.  F )  ->  A. a  e.  On  ( A. n  e.  F  E. b  e.  a  ( n `  b )  =/=  X  ->  |^| { x  e.  On  |  ( A `
 x )  =/= 
X }  e.  a ) )
38 elintrabg 3877 . . . 4  |-  ( |^| { x  e.  On  | 
( A `  x
)  =/=  X }  e.  On  ->  ( |^| { x  e.  On  | 
( A `  x
)  =/=  X }  e.  |^| { a  e.  On  |  A. n  e.  F  E. b  e.  a  ( n `  b )  =/=  X } 
<-> 
A. a  e.  On  ( A. n  e.  F  E. b  e.  a 
( n `  b
)  =/=  X  ->  |^| { x  e.  On  |  ( A `  x )  =/=  X }  e.  a )
) )
3938biimpar 471 . . 3  |-  ( (
|^| { x  e.  On  |  ( A `  x )  =/=  X }  e.  On  /\  A. a  e.  On  ( A. n  e.  F  E. b  e.  a 
( n `  b
)  =/=  X  ->  |^| { x  e.  On  |  ( A `  x )  =/=  X }  e.  a )
)  ->  |^| { x  e.  On  |  ( A `
 x )  =/= 
X }  e.  |^| { a  e.  On  |  A. n  e.  F  E. b  e.  a 
( n `  b
)  =/=  X }
)
4014, 37, 39syl2anc 642 . 2  |-  ( ( F  C_  No  /\  A  e.  F )  ->  |^| { x  e.  On  |  ( A `
 x )  =/= 
X }  e.  |^| { a  e.  On  |  A. n  e.  F  E. b  e.  a 
( n `  b
)  =/=  X }
)
41 nobndlem6.2 . 2  |-  C  = 
|^| { a  e.  On  |  A. n  e.  F  E. b  e.  a 
( n `  b
)  =/=  X }
4240, 41syl6eleqr 2376 1  |-  ( ( F  C_  No  /\  A  e.  F )  ->  |^| { x  e.  On  |  ( A `
 x )  =/= 
X }  e.  C
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1625    e. wcel 1686    =/= wne 2448   A.wral 2545   E.wrex 2546   {crab 2549    C_ wss 3154   (/)c0 3457   {cpr 3643   |^|cint 3864   Oncon0 4394   ` cfv 5257   1oc1o 6474   2oc2o 6475   Nocsur 24296   bdaycbday 24298
This theorem is referenced by:  nobndlem7  24354  nobndup  24356  nobnddown  24357
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-bday 24301
  Copyright terms: Public domain W3C validator