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

Theorem nobndlem6 27843
Description: Lemma for nobndup 27846 and nobnddown 27847. 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 27826 . . . . 5  |-  ( bday `  A )  e.  On
2 ssel2 3356 . . . . . 6  |-  ( ( F  C_  No  /\  A  e.  F )  ->  A  e.  No )
3 nobndlem6.1 . . . . . . . 8  |-  X  e. 
{ 1o ,  2o }
43nosgnn0i 27805 . . . . . . 7  |-  (/)  =/=  X
5 fvnobday 27828 . . . . . . . 8  |-  ( A  e.  No  ->  ( A `  ( bday `  A ) )  =  (/) )
65neeq1d 2626 . . . . . . 7  |-  ( A  e.  No  ->  (
( A `  ( bday `  A ) )  =/=  X  <->  (/)  =/=  X
) )
74, 6mpbiri 233 . . . . . 6  |-  ( A  e.  No  ->  ( A `  ( bday `  A ) )  =/= 
X )
82, 7syl 16 . . . . 5  |-  ( ( F  C_  No  /\  A  e.  F )  ->  ( A `  ( bday `  A ) )  =/= 
X )
9 fveq2 5696 . . . . . . 7  |-  ( x  =  ( bday `  A
)  ->  ( A `  x )  =  ( A `  ( bday `  A ) ) )
109neeq1d 2626 . . . . . 6  |-  ( x  =  ( bday `  A
)  ->  ( ( A `  x )  =/=  X  <->  ( A `  ( bday `  A )
)  =/=  X ) )
1110rspcev 3078 . . . . 5  |-  ( ( ( bday `  A
)  e.  On  /\  ( A `  ( bday `  A ) )  =/= 
X )  ->  E. x  e.  On  ( A `  x )  =/=  X
)
121, 8, 11sylancr 663 . . . 4  |-  ( ( F  C_  No  /\  A  e.  F )  ->  E. x  e.  On  ( A `  x )  =/=  X
)
13 onintrab2 6418 . . . 4  |-  ( E. x  e.  On  ( A `  x )  =/=  X  <->  |^| { x  e.  On  |  ( A `
 x )  =/= 
X }  e.  On )
1412, 13sylib 196 . . 3  |-  ( ( F  C_  No  /\  A  e.  F )  ->  |^| { x  e.  On  |  ( A `
 x )  =/= 
X }  e.  On )
15 fveq1 5695 . . . . . . . . 9  |-  ( n  =  A  ->  (
n `  b )  =  ( A `  b ) )
1615neeq1d 2626 . . . . . . . 8  |-  ( n  =  A  ->  (
( n `  b
)  =/=  X  <->  ( A `  b )  =/=  X
) )
1716rexbidv 2741 . . . . . . 7  |-  ( n  =  A  ->  ( E. b  e.  a 
( n `  b
)  =/=  X  <->  E. b  e.  a  ( A `  b )  =/=  X
) )
1817rspcv 3074 . . . . . 6  |-  ( A  e.  F  ->  ( A. n  e.  F  E. b  e.  a 
( n `  b
)  =/=  X  ->  E. b  e.  a 
( A `  b
)  =/=  X ) )
1918ad2antlr 726 . . . . 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 725 . . . . . . 7  |-  ( ( ( ( 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 754 . . . . . . 7  |-  ( ( ( ( F  C_  No  /\  A  e.  F
)  /\  a  e.  On )  /\  (
b  e.  a  /\  ( A `  b )  =/=  X ) )  ->  a  e.  On )
22 onelon 4749 . . . . . . . . . . 11  |-  ( ( a  e.  On  /\  b  e.  a )  ->  b  e.  On )
2322anim1i 568 . . . . . . . . . 10  |-  ( ( ( a  e.  On  /\  b  e.  a )  /\  ( A `  b )  =/=  X
)  ->  ( b  e.  On  /\  ( A `
 b )  =/= 
X ) )
2423anasss 647 . . . . . . . . 9  |-  ( ( a  e.  On  /\  ( b  e.  a  /\  ( A `  b )  =/=  X
) )  ->  (
b  e.  On  /\  ( A `  b )  =/=  X ) )
25 fveq2 5696 . . . . . . . . . . 11  |-  ( x  =  b  ->  ( A `  x )  =  ( A `  b ) )
2625neeq1d 2626 . . . . . . . . . 10  |-  ( x  =  b  ->  (
( A `  x
)  =/=  X  <->  ( A `  b )  =/=  X
) )
2726intminss 4159 . . . . . . . . 9  |-  ( ( b  e.  On  /\  ( A `  b )  =/=  X )  ->  |^| { x  e.  On  |  ( A `  x )  =/=  X }  C_  b )
2824, 27syl 16 . . . . . . . 8  |-  ( ( a  e.  On  /\  ( b  e.  a  /\  ( A `  b )  =/=  X
) )  ->  |^| { x  e.  On  |  ( A `
 x )  =/= 
X }  C_  b
)
2928adantll 713 . . . . . . 7  |-  ( ( ( ( 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 755 . . . . . . 7  |-  ( ( ( ( F  C_  No  /\  A  e.  F
)  /\  a  e.  On )  /\  (
b  e.  a  /\  ( A `  b )  =/=  X ) )  ->  b  e.  a )
31 ontr2 4771 . . . . . . . 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 ) )
3231imp 429 . . . . . . 7  |-  ( ( ( |^| { 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 1219 . . . . . 6  |-  ( ( ( ( F  C_  No  /\  A  e.  F
)  /\  a  e.  On )  /\  (
b  e.  a  /\  ( A `  b )  =/=  X ) )  ->  |^| { x  e.  On  |  ( A `
 x )  =/= 
X }  e.  a )
3433rexlimdvaa 2847 . . . . 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 ) )
3519, 34syld 44 . . . 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 ) )
3635ralrimiva 2804 . . 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 ) )
37 elintrabg 4146 . . . 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 )
) )
3837biimpar 485 . . 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 }
)
3914, 36, 38syl2anc 661 . 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 }
)
40 nobndlem6.2 . 2  |-  C  = 
|^| { a  e.  On  |  A. n  e.  F  E. b  e.  a 
( n `  b
)  =/=  X }
4139, 40syl6eleqr 2534 1  |-  ( ( F  C_  No  /\  A  e.  F )  ->  |^| { x  e.  On  |  ( A `
 x )  =/= 
X }  e.  C
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2611   A.wral 2720   E.wrex 2721   {crab 2724    C_ wss 3333   (/)c0 3642   {cpr 3884   |^|cint 4133   Oncon0 4724   ` cfv 5423   1oc1o 6918   2oc2o 6919   Nocsur 27786   bdaycbday 27788
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-1o 6925  df-2o 6926  df-no 27789  df-bday 27791
This theorem is referenced by:  nobndlem7  27844  nobndup  27846  nobnddown  27847
  Copyright terms: Public domain W3C validator