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Theorem nobndlem8 30600
Description: Lemma for nobndup 30601 and nobnddown 30602. 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 3056 . 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 30595 . . . 4  |-  ( ( F  C_  No  /\  F  e.  _V )  ->  ( bday `  ( C  X.  { S } ) )  =  C )
54, 3syl6eq 2503 . . 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 30593 . . . . 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 30580 . . . . . . . . . . 11  |-  bday  Fn  No
9 fnfvima 6148 . . . . . . . . . . 11  |-  ( (
bday  Fn  No  /\  F  C_  No  /\  n  e.  F )  ->  ( bday `  n )  e.  ( bday " F
) )
108, 9mp3an1 1353 . . . . . . . . . 10  |-  ( ( F  C_  No  /\  n  e.  F )  ->  ( bday `  n )  e.  ( bday " F
) )
11103adant2 1028 . . . . . . . . 9  |-  ( ( F  C_  No  /\  F  e.  _V  /\  n  e.  F )  ->  ( bday `  n )  e.  ( bday " F
) )
12 elssuni 4230 . . . . . . . . 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 30581 . . . . . . . . 9  |-  ( bday `  n )  e.  On
15 sucelon 6649 . . . . . . . . . . 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 1031 . . . . . . . . 9  |-  ( ( F  C_  No  /\  F  e.  _V  /\  n  e.  F )  ->  U. ( bday " F )  e.  On )
18 onsssuc 5513 . . . . . . . . 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 670 . . . . . . . 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 3429 . . . . . . . . 9  |-  ( ( F  C_  No  /\  n  e.  F )  ->  n  e.  No )
22 fvnobday 30583 . . . . . . . . . 10  |-  ( n  e.  No  ->  (
n `  ( bday `  n ) )  =  (/) )
232nosgnn0i 30558 . . . . . . . . . . 11  |-  (/)  =/=  S
2423a1i 11 . . . . . . . . . 10  |-  ( n  e.  No  ->  (/)  =/=  S
)
2522, 24eqnetrd 2693 . . . . . . . . 9  |-  ( n  e.  No  ->  (
n `  ( bday `  n ) )  =/= 
S )
2621, 25syl 17 . . . . . . . 8  |-  ( ( F  C_  No  /\  n  e.  F )  ->  (
n `  ( bday `  n ) )  =/= 
S )
27263adant2 1028 . . . . . . 7  |-  ( ( F  C_  No  /\  F  e.  _V  /\  n  e.  F )  ->  (
n `  ( bday `  n ) )  =/= 
S )
28 fveq2 5870 . . . . . . . . 9  |-  ( b  =  ( bday `  n
)  ->  ( n `  b )  =  ( n `  ( bday `  n ) ) )
2928neeq1d 2685 . . . . . . . 8  |-  ( b  =  ( bday `  n
)  ->  ( (
n `  b )  =/=  S  <->  ( n `  ( bday `  n )
)  =/=  S ) )
3029rspcev 3152 . . . . . . 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 667 . . . . . 6  |-  ( ( F  C_  No  /\  F  e.  _V  /\  n  e.  F )  ->  E. b  e.  suc  U. ( bday " F ) ( n `
 b )  =/= 
S )
32313expa 1209 . . . . 5  |-  ( ( ( F  C_  No  /\  F  e.  _V )  /\  n  e.  F
)  ->  E. b  e.  suc  U. ( bday " F ) ( n `
 b )  =/= 
S )
3332ralrimiva 2804 . . . 4  |-  ( ( F  C_  No  /\  F  e.  _V )  ->  A. n  e.  F  E. b  e.  suc  U. ( bday " F ) ( n `
 b )  =/= 
S )
34 rexeq 2990 . . . . . 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 4264 . . . 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 667 . . 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 3468 . 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 986    = wceq 1446    e. wcel 1889    =/= wne 2624   A.wral 2739   E.wrex 2740   {crab 2743   _Vcvv 3047    C_ wss 3406   (/)c0 3733   {csn 3970   {cpr 3972   U.cuni 4201   |^|cint 4237    X. cxp 4835   "cima 4840   Oncon0 5426   suc csuc 5428    Fn wfn 5580   ` cfv 5585   1oc1o 7180   2oc2o 7181   Nocsur 30539   bdaycbday 30541
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-reu 2746  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-int 4238  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-we 4798  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-ord 5429  df-on 5430  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-1o 7187  df-2o 7188  df-no 30542  df-bday 30544
This theorem is referenced by:  nobndup  30601  nobnddown  30602
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