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Theorem nobndlem6 27767
Description: Lemma for nobndup 27770 and nobnddown 27771. 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 27750 . . . . 5  |-  ( bday `  A )  e.  On
2 ssel2 3348 . . . . . 6  |-  ( ( F  C_  No  /\  A  e.  F )  ->  A  e.  No )
3 nobndlem6.1 . . . . . . . 8  |-  X  e. 
{ 1o ,  2o }
43nosgnn0i 27729 . . . . . . 7  |-  (/)  =/=  X
5 fvnobday 27752 . . . . . . . 8  |-  ( A  e.  No  ->  ( A `  ( bday `  A ) )  =  (/) )
65neeq1d 2619 . . . . . . 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 5688 . . . . . . 7  |-  ( x  =  ( bday `  A
)  ->  ( A `  x )  =  ( A `  ( bday `  A ) ) )
109neeq1d 2619 . . . . . 6  |-  ( x  =  ( bday `  A
)  ->  ( ( A `  x )  =/=  X  <->  ( A `  ( bday `  A )
)  =/=  X ) )
1110rspcev 3070 . . . . 5  |-  ( ( ( bday `  A
)  e.  On  /\  ( A `  ( bday `  A ) )  =/= 
X )  ->  E. x  e.  On  ( A `  x )  =/=  X
)
121, 8, 11sylancr 658 . . . 4  |-  ( ( F  C_  No  /\  A  e.  F )  ->  E. x  e.  On  ( A `  x )  =/=  X
)
13 onintrab2 6412 . . . 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 5687 . . . . . . . . 9  |-  ( n  =  A  ->  (
n `  b )  =  ( A `  b ) )
1615neeq1d 2619 . . . . . . . 8  |-  ( n  =  A  ->  (
( n `  b
)  =/=  X  <->  ( A `  b )  =/=  X
) )
1716rexbidv 2734 . . . . . . 7  |-  ( n  =  A  ->  ( E. b  e.  a 
( n `  b
)  =/=  X  <->  E. b  e.  a  ( A `  b )  =/=  X
) )
1817rspcv 3066 . . . . . 6  |-  ( A  e.  F  ->  ( A. n  e.  F  E. b  e.  a 
( n `  b
)  =/=  X  ->  E. b  e.  a 
( A `  b
)  =/=  X ) )
1918ad2antlr 721 . . . . 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 720 . . . . . . 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 749 . . . . . . 7  |-  ( ( ( ( F  C_  No  /\  A  e.  F
)  /\  a  e.  On )  /\  (
b  e.  a  /\  ( A `  b )  =/=  X ) )  ->  a  e.  On )
22 onelon 4740 . . . . . . . . . . 11  |-  ( ( a  e.  On  /\  b  e.  a )  ->  b  e.  On )
2322anim1i 565 . . . . . . . . . 10  |-  ( ( ( a  e.  On  /\  b  e.  a )  /\  ( A `  b )  =/=  X
)  ->  ( b  e.  On  /\  ( A `
 b )  =/= 
X ) )
2423anasss 642 . . . . . . . . 9  |-  ( ( a  e.  On  /\  ( b  e.  a  /\  ( A `  b )  =/=  X
) )  ->  (
b  e.  On  /\  ( A `  b )  =/=  X ) )
25 fveq2 5688 . . . . . . . . . . 11  |-  ( x  =  b  ->  ( A `  x )  =  ( A `  b ) )
2625neeq1d 2619 . . . . . . . . . 10  |-  ( x  =  b  ->  (
( A `  x
)  =/=  X  <->  ( A `  b )  =/=  X
) )
2726intminss 4151 . . . . . . . . 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 708 . . . . . . 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 750 . . . . . . 7  |-  ( ( ( ( F  C_  No  /\  A  e.  F
)  /\  a  e.  On )  /\  (
b  e.  a  /\  ( A `  b )  =/=  X ) )  ->  b  e.  a )
31 ontr2 4762 . . . . . . . 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 1214 . . . . . 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 2840 . . . . 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 2797 . . 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 4138 . . . 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 482 . . 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 656 . 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 2532 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 1364    e. wcel 1761    =/= wne 2604   A.wral 2713   E.wrex 2714   {crab 2717    C_ wss 3325   (/)c0 3634   {cpr 3876   |^|cint 4125   Oncon0 4715   ` cfv 5415   1oc1o 6909   2oc2o 6910   Nocsur 27710   bdaycbday 27712
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-1o 6916  df-2o 6917  df-no 27713  df-bday 27715
This theorem is referenced by:  nobndlem7  27768  nobndup  27770  nobnddown  27771
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