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Theorem nobndlem8 30591
Description: Lemma for nobndup 30592 and nobnddown 30593. Bound the birthday of  ( C  X.  { S } ) above. (Contributed by Scott Fenton, 10-Apr-2017.)
Hypotheses
Ref Expression
nobndlem8.1  |-  S  e. 
{ 1o ,  2o }
nobndlem8.2  |-  C  = 
|^| { a  e.  On  |  A. n  e.  F  E. b  e.  a 
( n `  b
)  =/=  S }
Assertion
Ref Expression
nobndlem8  |-  ( ( F  C_  No  /\  F  e.  A )  ->  ( bday `  ( C  X.  { S } ) ) 
C_  suc  U. ( bday " F ) )
Distinct variable groups:    F, a,
b, n    S, a,
b
Allowed substitution hints:    A( n, a, b)    C( n, a, b)    S( n)

Proof of Theorem nobndlem8
StepHypRef Expression
1 elex 3091 . 2  |-  ( F  e.  A  ->  F  e.  _V )
2 nobndlem8.1 . . . . 5  |-  S  e. 
{ 1o ,  2o }
3 nobndlem8.2 . . . . 5  |-  C  = 
|^| { a  e.  On  |  A. n  e.  F  E. b  e.  a 
( n `  b
)  =/=  S }
42, 3nobndlem3 30586 . . . 4  |-  ( ( F  C_  No  /\  F  e.  _V )  ->  ( bday `  ( C  X.  { S } ) )  =  C )
54, 3syl6eq 2480 . . 3  |-  ( ( F  C_  No  /\  F  e.  _V )  ->  ( bday `  ( C  X.  { S } ) )  =  |^| { a  e.  On  |  A. n  e.  F  E. b  e.  a  (
n `  b )  =/=  S } )
6 nobndlem1 30584 . . . . 5  |-  ( F  e.  _V  ->  suc  U. ( bday " F
)  e.  On )
76adantl 468 . . . 4  |-  ( ( F  C_  No  /\  F  e.  _V )  ->  suc  U. ( bday " F
)  e.  On )
8 bdayfn 30571 . . . . . . . . . . 11  |-  bday  Fn  No
9 fnfvima 6157 . . . . . . . . . . 11  |-  ( (
bday  Fn  No  /\  F  C_  No  /\  n  e.  F )  ->  ( bday `  n )  e.  ( bday " F
) )
108, 9mp3an1 1348 . . . . . . . . . 10  |-  ( ( F  C_  No  /\  n  e.  F )  ->  ( bday `  n )  e.  ( bday " F
) )
11103adant2 1025 . . . . . . . . 9  |-  ( ( F  C_  No  /\  F  e.  _V  /\  n  e.  F )  ->  ( bday `  n )  e.  ( bday " F
) )
12 elssuni 4247 . . . . . . . . 9  |-  ( (
bday `  n )  e.  ( bday " F
)  ->  ( bday `  n )  C_  U. ( bday " F ) )
1311, 12syl 17 . . . . . . . 8  |-  ( ( F  C_  No  /\  F  e.  _V  /\  n  e.  F )  ->  ( bday `  n )  C_  U. ( bday " F
) )
14 bdayelon 30572 . . . . . . . . 9  |-  ( bday `  n )  e.  On
15 sucelon 6657 . . . . . . . . . . 11  |-  ( U. ( bday " F )  e.  On  <->  suc  U. ( bday " F )  e.  On )
166, 15sylibr 216 . . . . . . . . . 10  |-  ( F  e.  _V  ->  U. ( bday " F )  e.  On )
17163ad2ant2 1028 . . . . . . . . 9  |-  ( ( F  C_  No  /\  F  e.  _V  /\  n  e.  F )  ->  U. ( bday " F )  e.  On )
18 onsssuc 5528 . . . . . . . . 9  |-  ( ( ( bday `  n
)  e.  On  /\  U. ( bday " F
)  e.  On )  ->  ( ( bday `  n )  C_  U. ( bday " F )  <->  ( bday `  n )  e.  suc  U. ( bday " F
) ) )
1914, 17, 18sylancr 668 . . . . . . . 8  |-  ( ( F  C_  No  /\  F  e.  _V  /\  n  e.  F )  ->  (
( bday `  n )  C_ 
U. ( bday " F
)  <->  ( bday `  n
)  e.  suc  U. ( bday " F ) ) )
2013, 19mpbid 214 . . . . . . 7  |-  ( ( F  C_  No  /\  F  e.  _V  /\  n  e.  F )  ->  ( bday `  n )  e. 
suc  U. ( bday " F
) )
21 ssel2 3461 . . . . . . . . 9  |-  ( ( F  C_  No  /\  n  e.  F )  ->  n  e.  No )
22 fvnobday 30574 . . . . . . . . . 10  |-  ( n  e.  No  ->  (
n `  ( bday `  n ) )  =  (/) )
232nosgnn0i 30551 . . . . . . . . . . 11  |-  (/)  =/=  S
2423a1i 11 . . . . . . . . . 10  |-  ( n  e.  No  ->  (/)  =/=  S
)
2522, 24eqnetrd 2718 . . . . . . . . 9  |-  ( n  e.  No  ->  (
n `  ( bday `  n ) )  =/= 
S )
2621, 25syl 17 . . . . . . . 8  |-  ( ( F  C_  No  /\  n  e.  F )  ->  (
n `  ( bday `  n ) )  =/= 
S )
27263adant2 1025 . . . . . . 7  |-  ( ( F  C_  No  /\  F  e.  _V  /\  n  e.  F )  ->  (
n `  ( bday `  n ) )  =/= 
S )
28 fveq2 5880 . . . . . . . . 9  |-  ( b  =  ( bday `  n
)  ->  ( n `  b )  =  ( n `  ( bday `  n ) ) )
2928neeq1d 2702 . . . . . . . 8  |-  ( b  =  ( bday `  n
)  ->  ( (
n `  b )  =/=  S  <->  ( n `  ( bday `  n )
)  =/=  S ) )
3029rspcev 3183 . . . . . . 7  |-  ( ( ( bday `  n
)  e.  suc  U. ( bday " F )  /\  ( n `  ( bday `  n )
)  =/=  S )  ->  E. b  e.  suc  U. ( bday " F
) ( n `  b )  =/=  S
)
3120, 27, 30syl2anc 666 . . . . . 6  |-  ( ( F  C_  No  /\  F  e.  _V  /\  n  e.  F )  ->  E. b  e.  suc  U. ( bday " F ) ( n `
 b )  =/= 
S )
32313expa 1206 . . . . 5  |-  ( ( ( F  C_  No  /\  F  e.  _V )  /\  n  e.  F
)  ->  E. b  e.  suc  U. ( bday " F ) ( n `
 b )  =/= 
S )
3332ralrimiva 2840 . . . 4  |-  ( ( F  C_  No  /\  F  e.  _V )  ->  A. n  e.  F  E. b  e.  suc  U. ( bday " F ) ( n `
 b )  =/= 
S )
34 rexeq 3027 . . . . . 6  |-  ( a  =  suc  U. ( bday " F )  -> 
( E. b  e.  a  ( n `  b )  =/=  S  <->  E. b  e.  suc  U. ( bday " F ) ( n `  b
)  =/=  S ) )
3534ralbidv 2865 . . . . 5  |-  ( a  =  suc  U. ( bday " F )  -> 
( A. n  e.  F  E. b  e.  a  ( n `  b )  =/=  S  <->  A. n  e.  F  E. b  e.  suc  U. ( bday " F ) ( n `  b )  =/=  S ) )
3635intminss 4281 . . . 4  |-  ( ( suc  U. ( bday " F )  e.  On  /\ 
A. n  e.  F  E. b  e.  suc  U. ( bday " F
) ( n `  b )  =/=  S
)  ->  |^| { a  e.  On  |  A. n  e.  F  E. b  e.  a  (
n `  b )  =/=  S }  C_  suc  U. ( bday " F
) )
377, 33, 36syl2anc 666 . . 3  |-  ( ( F  C_  No  /\  F  e.  _V )  ->  |^| { a  e.  On  |  A. n  e.  F  E. b  e.  a  (
n `  b )  =/=  S }  C_  suc  U. ( bday " F
) )
385, 37eqsstrd 3500 . 2  |-  ( ( F  C_  No  /\  F  e.  _V )  ->  ( bday `  ( C  X.  { S } ) ) 
C_  suc  U. ( bday " F ) )
391, 38sylan2 477 1  |-  ( ( F  C_  No  /\  F  e.  A )  ->  ( bday `  ( C  X.  { S } ) ) 
C_  suc  U. ( bday " F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 983    = wceq 1438    e. wcel 1869    =/= wne 2619   A.wral 2776   E.wrex 2777   {crab 2780   _Vcvv 3082    C_ wss 3438   (/)c0 3763   {csn 3998   {cpr 4000   U.cuni 4218   |^|cint 4254    X. cxp 4850   "cima 4855   Oncon0 5441   suc csuc 5443    Fn wfn 5595   ` cfv 5600   1oc1o 7185   2oc2o 7186   Nocsur 30532   bdaycbday 30534
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-rep 4535  ax-sep 4545  ax-nul 4554  ax-pow 4601  ax-pr 4659  ax-un 6596
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-reu 2783  df-rab 2785  df-v 3084  df-sbc 3302  df-csb 3398  df-dif 3441  df-un 3443  df-in 3445  df-ss 3452  df-pss 3454  df-nul 3764  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4219  df-int 4255  df-iun 4300  df-br 4423  df-opab 4482  df-mpt 4483  df-tr 4518  df-eprel 4763  df-id 4767  df-po 4773  df-so 4774  df-fr 4811  df-we 4813  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-ord 5444  df-on 5445  df-suc 5447  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-fv 5608  df-1o 7192  df-2o 7193  df-no 30535  df-bday 30537
This theorem is referenced by:  nobndup  30592  nobnddown  30593
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