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Theorem nobndlem8 27960
Description: Lemma for nobndup 27961 and nobnddown 27962. 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 3063 . 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 27955 . . . 4  |-  ( ( F  C_  No  /\  F  e.  _V )  ->  ( bday `  ( C  X.  { S } ) )  =  C )
54, 3syl6eq 2506 . . 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 27953 . . . . 5  |-  ( F  e.  _V  ->  suc  U. ( bday " F
)  e.  On )
76adantl 466 . . . 4  |-  ( ( F  C_  No  /\  F  e.  _V )  ->  suc  U. ( bday " F
)  e.  On )
8 bdayfn 27940 . . . . . . . . . . 11  |-  bday  Fn  No
9 fnfvima 6040 . . . . . . . . . . 11  |-  ( (
bday  Fn  No  /\  F  C_  No  /\  n  e.  F )  ->  ( bday `  n )  e.  ( bday " F
) )
108, 9mp3an1 1302 . . . . . . . . . 10  |-  ( ( F  C_  No  /\  n  e.  F )  ->  ( bday `  n )  e.  ( bday " F
) )
11103adant2 1007 . . . . . . . . 9  |-  ( ( F  C_  No  /\  F  e.  _V  /\  n  e.  F )  ->  ( bday `  n )  e.  ( bday " F
) )
12 elssuni 4205 . . . . . . . . 9  |-  ( (
bday `  n )  e.  ( bday " F
)  ->  ( bday `  n )  C_  U. ( bday " F ) )
1311, 12syl 16 . . . . . . . 8  |-  ( ( F  C_  No  /\  F  e.  _V  /\  n  e.  F )  ->  ( bday `  n )  C_  U. ( bday " F
) )
14 bdayelon 27941 . . . . . . . . 9  |-  ( bday `  n )  e.  On
15 sucelon 6514 . . . . . . . . . . 11  |-  ( U. ( bday " F )  e.  On  <->  suc  U. ( bday " F )  e.  On )
166, 15sylibr 212 . . . . . . . . . 10  |-  ( F  e.  _V  ->  U. ( bday " F )  e.  On )
17163ad2ant2 1010 . . . . . . . . 9  |-  ( ( F  C_  No  /\  F  e.  _V  /\  n  e.  F )  ->  U. ( bday " F )  e.  On )
18 onsssuc 4890 . . . . . . . . 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 663 . . . . . . . 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 210 . . . . . . 7  |-  ( ( F  C_  No  /\  F  e.  _V  /\  n  e.  F )  ->  ( bday `  n )  e. 
suc  U. ( bday " F
) )
21 ssel2 3435 . . . . . . . . 9  |-  ( ( F  C_  No  /\  n  e.  F )  ->  n  e.  No )
22 fvnobday 27943 . . . . . . . . . 10  |-  ( n  e.  No  ->  (
n `  ( bday `  n ) )  =  (/) )
232nosgnn0i 27920 . . . . . . . . . . 11  |-  (/)  =/=  S
2423a1i 11 . . . . . . . . . 10  |-  ( n  e.  No  ->  (/)  =/=  S
)
2522, 24eqnetrd 2738 . . . . . . . . 9  |-  ( n  e.  No  ->  (
n `  ( bday `  n ) )  =/= 
S )
2621, 25syl 16 . . . . . . . 8  |-  ( ( F  C_  No  /\  n  e.  F )  ->  (
n `  ( bday `  n ) )  =/= 
S )
27263adant2 1007 . . . . . . 7  |-  ( ( F  C_  No  /\  F  e.  _V  /\  n  e.  F )  ->  (
n `  ( bday `  n ) )  =/= 
S )
28 fveq2 5775 . . . . . . . . 9  |-  ( b  =  ( bday `  n
)  ->  ( n `  b )  =  ( n `  ( bday `  n ) ) )
2928neeq1d 2722 . . . . . . . 8  |-  ( b  =  ( bday `  n
)  ->  ( (
n `  b )  =/=  S  <->  ( n `  ( bday `  n )
)  =/=  S ) )
3029rspcev 3155 . . . . . . 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 661 . . . . . 6  |-  ( ( F  C_  No  /\  F  e.  _V  /\  n  e.  F )  ->  E. b  e.  suc  U. ( bday " F ) ( n `
 b )  =/= 
S )
32313expa 1188 . . . . 5  |-  ( ( ( F  C_  No  /\  F  e.  _V )  /\  n  e.  F
)  ->  E. b  e.  suc  U. ( bday " F ) ( n `
 b )  =/= 
S )
3332ralrimiva 2881 . . . 4  |-  ( ( F  C_  No  /\  F  e.  _V )  ->  A. n  e.  F  E. b  e.  suc  U. ( bday " F ) ( n `
 b )  =/= 
S )
34 rexeq 3000 . . . . . 6  |-  ( a  =  suc  U. ( bday " F )  -> 
( E. b  e.  a  ( n `  b )  =/=  S  <->  E. b  e.  suc  U. ( bday " F ) ( n `  b
)  =/=  S ) )
3534ralbidv 2815 . . . . 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 4238 . . . 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 661 . . 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 3474 . 2  |-  ( ( F  C_  No  /\  F  e.  _V )  ->  ( bday `  ( C  X.  { S } ) ) 
C_  suc  U. ( bday " F ) )
391, 38sylan2 474 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 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1757    =/= wne 2641   A.wral 2792   E.wrex 2793   {crab 2796   _Vcvv 3054    C_ wss 3412   (/)c0 3721   {csn 3961   {cpr 3963   U.cuni 4175   |^|cint 4212   Oncon0 4803   suc csuc 4805    X. cxp 4922   "cima 4927    Fn wfn 5497   ` cfv 5502   1oc1o 6999   2oc2o 7000   Nocsur 27901   bdaycbday 27903
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-rep 4487  ax-sep 4497  ax-nul 4505  ax-pow 4554  ax-pr 4615  ax-un 6458
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-eu 2263  df-mo 2264  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-ral 2797  df-rex 2798  df-reu 2799  df-rab 2801  df-v 3056  df-sbc 3271  df-csb 3373  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-pss 3428  df-nul 3722  df-if 3876  df-pw 3946  df-sn 3962  df-pr 3964  df-tp 3966  df-op 3968  df-uni 4176  df-int 4213  df-iun 4257  df-br 4377  df-opab 4435  df-mpt 4436  df-tr 4470  df-eprel 4716  df-id 4720  df-po 4725  df-so 4726  df-fr 4763  df-we 4765  df-ord 4806  df-on 4807  df-suc 4809  df-xp 4930  df-rel 4931  df-cnv 4932  df-co 4933  df-dm 4934  df-rn 4935  df-res 4936  df-ima 4937  df-iota 5465  df-fun 5504  df-fn 5505  df-f 5506  df-f1 5507  df-fo 5508  df-f1o 5509  df-fv 5510  df-1o 7006  df-2o 7007  df-no 27904  df-bday 27906
This theorem is referenced by:  nobndup  27961  nobnddown  27962
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