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Theorem nobndlem6 25652
Description: Lemma for nobndup 25655 and nobnddown 25656. 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 25635 . . . . 5  |-  ( bday `  A )  e.  On
2 ssel2 3343 . . . . . 6  |-  ( ( F  C_  No  /\  A  e.  F )  ->  A  e.  No )
3 nobndlem6.1 . . . . . . . 8  |-  X  e. 
{ 1o ,  2o }
43nosgnn0i 25614 . . . . . . 7  |-  (/)  =/=  X
5 fvnobday 25637 . . . . . . . 8  |-  ( A  e.  No  ->  ( A `  ( bday `  A ) )  =  (/) )
65neeq1d 2614 . . . . . . 7  |-  ( A  e.  No  ->  (
( A `  ( bday `  A ) )  =/=  X  <->  (/)  =/=  X
) )
74, 6mpbiri 225 . . . . . 6  |-  ( A  e.  No  ->  ( A `  ( bday `  A ) )  =/= 
X )
82, 7syl 16 . . . . 5  |-  ( ( F  C_  No  /\  A  e.  F )  ->  ( A `  ( bday `  A ) )  =/= 
X )
9 fveq2 5728 . . . . . . 7  |-  ( x  =  ( bday `  A
)  ->  ( A `  x )  =  ( A `  ( bday `  A ) ) )
109neeq1d 2614 . . . . . 6  |-  ( x  =  ( bday `  A
)  ->  ( ( A `  x )  =/=  X  <->  ( A `  ( bday `  A )
)  =/=  X ) )
1110rspcev 3052 . . . . 5  |-  ( ( ( bday `  A
)  e.  On  /\  ( A `  ( bday `  A ) )  =/= 
X )  ->  E. x  e.  On  ( A `  x )  =/=  X
)
121, 8, 11sylancr 645 . . . 4  |-  ( ( F  C_  No  /\  A  e.  F )  ->  E. x  e.  On  ( A `  x )  =/=  X
)
13 onintrab2 4782 . . . 4  |-  ( E. x  e.  On  ( A `  x )  =/=  X  <->  |^| { x  e.  On  |  ( A `
 x )  =/= 
X }  e.  On )
1412, 13sylib 189 . . 3  |-  ( ( F  C_  No  /\  A  e.  F )  ->  |^| { x  e.  On  |  ( A `
 x )  =/= 
X }  e.  On )
15 fveq1 5727 . . . . . . . . 9  |-  ( n  =  A  ->  (
n `  b )  =  ( A `  b ) )
1615neeq1d 2614 . . . . . . . 8  |-  ( n  =  A  ->  (
( n `  b
)  =/=  X  <->  ( A `  b )  =/=  X
) )
1716rexbidv 2726 . . . . . . 7  |-  ( n  =  A  ->  ( E. b  e.  a 
( n `  b
)  =/=  X  <->  E. b  e.  a  ( A `  b )  =/=  X
) )
1817rspcv 3048 . . . . . 6  |-  ( A  e.  F  ->  ( A. n  e.  F  E. b  e.  a 
( n `  b
)  =/=  X  ->  E. b  e.  a 
( A `  b
)  =/=  X ) )
1918ad2antlr 708 . . . . 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 707 . . . . . . 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 732 . . . . . . 7  |-  ( ( ( ( F  C_  No  /\  A  e.  F
)  /\  a  e.  On )  /\  (
b  e.  a  /\  ( A `  b )  =/=  X ) )  ->  a  e.  On )
22 onelon 4606 . . . . . . . . . . 11  |-  ( ( a  e.  On  /\  b  e.  a )  ->  b  e.  On )
2322anim1i 552 . . . . . . . . . 10  |-  ( ( ( a  e.  On  /\  b  e.  a )  /\  ( A `  b )  =/=  X
)  ->  ( b  e.  On  /\  ( A `
 b )  =/= 
X ) )
2423anasss 629 . . . . . . . . 9  |-  ( ( a  e.  On  /\  ( b  e.  a  /\  ( A `  b )  =/=  X
) )  ->  (
b  e.  On  /\  ( A `  b )  =/=  X ) )
25 fveq2 5728 . . . . . . . . . . 11  |-  ( x  =  b  ->  ( A `  x )  =  ( A `  b ) )
2625neeq1d 2614 . . . . . . . . . 10  |-  ( x  =  b  ->  (
( A `  x
)  =/=  X  <->  ( A `  b )  =/=  X
) )
2726intminss 4076 . . . . . . . . 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 695 . . . . . . 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 733 . . . . . . 7  |-  ( ( ( ( F  C_  No  /\  A  e.  F
)  /\  a  e.  On )  /\  (
b  e.  a  /\  ( A `  b )  =/=  X ) )  ->  b  e.  a )
31 ontr2 4628 . . . . . . . 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 419 . . . . . . 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 1185 . . . . . 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 2831 . . . . 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 42 . . . 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 2789 . . 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 4063 . . . 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 472 . . 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 643 . 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 2527 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 359    = wceq 1652    e. wcel 1725    =/= wne 2599   A.wral 2705   E.wrex 2706   {crab 2709    C_ wss 3320   (/)c0 3628   {cpr 3815   |^|cint 4050   Oncon0 4581   ` cfv 5454   1oc1o 6717   2oc2o 6718   Nocsur 25595   bdaycbday 25597
This theorem is referenced by:  nobndlem7  25653  nobndup  25655  nobnddown  25656
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-suc 4587  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-1o 6724  df-2o 6725  df-no 25598  df-bday 25600
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