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Theorem nobndlem6 25565
Description: Lemma for nobndup 25568 and nobnddown 25569. 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 25548 . . . . 5  |-  ( bday `  A )  e.  On
2 ssel2 3303 . . . . . 6  |-  ( ( F  C_  No  /\  A  e.  F )  ->  A  e.  No )
3 nobndlem6.1 . . . . . . . 8  |-  X  e. 
{ 1o ,  2o }
43nosgnn0i 25527 . . . . . . 7  |-  (/)  =/=  X
5 fvnobday 25550 . . . . . . . 8  |-  ( A  e.  No  ->  ( A `  ( bday `  A ) )  =  (/) )
65neeq1d 2580 . . . . . . 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 5687 . . . . . . 7  |-  ( x  =  ( bday `  A
)  ->  ( A `  x )  =  ( A `  ( bday `  A ) ) )
109neeq1d 2580 . . . . . 6  |-  ( x  =  ( bday `  A
)  ->  ( ( A `  x )  =/=  X  <->  ( A `  ( bday `  A )
)  =/=  X ) )
1110rspcev 3012 . . . . 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 4741 . . . 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 5686 . . . . . . . . 9  |-  ( n  =  A  ->  (
n `  b )  =  ( A `  b ) )
1615neeq1d 2580 . . . . . . . 8  |-  ( n  =  A  ->  (
( n `  b
)  =/=  X  <->  ( A `  b )  =/=  X
) )
1716rexbidv 2687 . . . . . . 7  |-  ( n  =  A  ->  ( E. b  e.  a 
( n `  b
)  =/=  X  <->  E. b  e.  a  ( A `  b )  =/=  X
) )
1817rspcv 3008 . . . . . 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 4566 . . . . . . . . . . 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 5687 . . . . . . . . . . 11  |-  ( x  =  b  ->  ( A `  x )  =  ( A `  b ) )
2625neeq1d 2580 . . . . . . . . . 10  |-  ( x  =  b  ->  (
( A `  x
)  =/=  X  <->  ( A `  b )  =/=  X
) )
2726intminss 4036 . . . . . . . . 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 4588 . . . . . . . 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 2791 . . . . 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 2749 . . 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 4023 . . . 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 2495 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 1649    e. wcel 1721    =/= wne 2567   A.wral 2666   E.wrex 2667   {crab 2670    C_ wss 3280   (/)c0 3588   {cpr 3775   |^|cint 4010   Oncon0 4541   ` cfv 5413   1oc1o 6676   2oc2o 6677   Nocsur 25508   bdaycbday 25510
This theorem is referenced by:  nobndlem7  25566  nobndup  25568  nobnddown  25569
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-suc 4547  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-1o 6683  df-2o 6684  df-no 25511  df-bday 25513
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