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Theorem nobndlem8 30659
Description: Lemma for nobndup 30660 and nobnddown 30661. 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 3040 . 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 30654 . . . 4  |-  ( ( F  C_  No  /\  F  e.  _V )  ->  ( bday `  ( C  X.  { S } ) )  =  C )
54, 3syl6eq 2521 . . 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 30652 . . . . 5  |-  ( F  e.  _V  ->  suc  U. ( bday " F
)  e.  On )
76adantl 473 . . . 4  |-  ( ( F  C_  No  /\  F  e.  _V )  ->  suc  U. ( bday " F
)  e.  On )
8 bdayfn 30639 . . . . . . . . . . 11  |-  bday  Fn  No
9 fnfvima 6161 . . . . . . . . . . 11  |-  ( (
bday  Fn  No  /\  F  C_  No  /\  n  e.  F )  ->  ( bday `  n )  e.  ( bday " F
) )
108, 9mp3an1 1377 . . . . . . . . . 10  |-  ( ( F  C_  No  /\  n  e.  F )  ->  ( bday `  n )  e.  ( bday " F
) )
11103adant2 1049 . . . . . . . . 9  |-  ( ( F  C_  No  /\  F  e.  _V  /\  n  e.  F )  ->  ( bday `  n )  e.  ( bday " F
) )
12 elssuni 4219 . . . . . . . . 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 30640 . . . . . . . . 9  |-  ( bday `  n )  e.  On
15 sucelon 6663 . . . . . . . . . . 11  |-  ( U. ( bday " F )  e.  On  <->  suc  U. ( bday " F )  e.  On )
166, 15sylibr 217 . . . . . . . . . 10  |-  ( F  e.  _V  ->  U. ( bday " F )  e.  On )
17163ad2ant2 1052 . . . . . . . . 9  |-  ( ( F  C_  No  /\  F  e.  _V  /\  n  e.  F )  ->  U. ( bday " F )  e.  On )
18 onsssuc 5517 . . . . . . . . 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 676 . . . . . . . 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 215 . . . . . . 7  |-  ( ( F  C_  No  /\  F  e.  _V  /\  n  e.  F )  ->  ( bday `  n )  e. 
suc  U. ( bday " F
) )
21 ssel2 3413 . . . . . . . . 9  |-  ( ( F  C_  No  /\  n  e.  F )  ->  n  e.  No )
22 fvnobday 30642 . . . . . . . . . 10  |-  ( n  e.  No  ->  (
n `  ( bday `  n ) )  =  (/) )
232nosgnn0i 30617 . . . . . . . . . . 11  |-  (/)  =/=  S
2423a1i 11 . . . . . . . . . 10  |-  ( n  e.  No  ->  (/)  =/=  S
)
2522, 24eqnetrd 2710 . . . . . . . . 9  |-  ( n  e.  No  ->  (
n `  ( bday `  n ) )  =/= 
S )
2621, 25syl 17 . . . . . . . 8  |-  ( ( F  C_  No  /\  n  e.  F )  ->  (
n `  ( bday `  n ) )  =/= 
S )
27263adant2 1049 . . . . . . 7  |-  ( ( F  C_  No  /\  F  e.  _V  /\  n  e.  F )  ->  (
n `  ( bday `  n ) )  =/= 
S )
28 fveq2 5879 . . . . . . . . 9  |-  ( b  =  ( bday `  n
)  ->  ( n `  b )  =  ( n `  ( bday `  n ) ) )
2928neeq1d 2702 . . . . . . . 8  |-  ( b  =  ( bday `  n
)  ->  ( (
n `  b )  =/=  S  <->  ( n `  ( bday `  n )
)  =/=  S ) )
3029rspcev 3136 . . . . . . 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 673 . . . . . 6  |-  ( ( F  C_  No  /\  F  e.  _V  /\  n  e.  F )  ->  E. b  e.  suc  U. ( bday " F ) ( n `
 b )  =/= 
S )
32313expa 1231 . . . . 5  |-  ( ( ( F  C_  No  /\  F  e.  _V )  /\  n  e.  F
)  ->  E. b  e.  suc  U. ( bday " F ) ( n `
 b )  =/= 
S )
3332ralrimiva 2809 . . . 4  |-  ( ( F  C_  No  /\  F  e.  _V )  ->  A. n  e.  F  E. b  e.  suc  U. ( bday " F ) ( n `
 b )  =/= 
S )
34 rexeq 2974 . . . . . 6  |-  ( a  =  suc  U. ( bday " F )  -> 
( E. b  e.  a  ( n `  b )  =/=  S  <->  E. b  e.  suc  U. ( bday " F ) ( n `  b
)  =/=  S ) )
3534ralbidv 2829 . . . . 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 4252 . . . 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 673 . . 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 3452 . 2  |-  ( ( F  C_  No  /\  F  e.  _V )  ->  ( bday `  ( C  X.  { S } ) ) 
C_  suc  U. ( bday " F ) )
391, 38sylan2 482 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 189    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904    =/= wne 2641   A.wral 2756   E.wrex 2757   {crab 2760   _Vcvv 3031    C_ wss 3390   (/)c0 3722   {csn 3959   {cpr 3961   U.cuni 4190   |^|cint 4226    X. cxp 4837   "cima 4842   Oncon0 5430   suc csuc 5432    Fn wfn 5584   ` cfv 5589   1oc1o 7193   2oc2o 7194   Nocsur 30598   bdaycbday 30600
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-ord 5433  df-on 5434  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-1o 7200  df-2o 7201  df-no 30601  df-bday 30603
This theorem is referenced by:  nobndup  30660  nobnddown  30661
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