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Theorem nobndlem8 29624
Description: Lemma for nobndup 29625 and nobnddown 29626. 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 3043 . 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 29619 . . . 4  |-  ( ( F  C_  No  /\  F  e.  _V )  ->  ( bday `  ( C  X.  { S } ) )  =  C )
54, 3syl6eq 2439 . . 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 29617 . . . . 5  |-  ( F  e.  _V  ->  suc  U. ( bday " F
)  e.  On )
76adantl 464 . . . 4  |-  ( ( F  C_  No  /\  F  e.  _V )  ->  suc  U. ( bday " F
)  e.  On )
8 bdayfn 29604 . . . . . . . . . . 11  |-  bday  Fn  No
9 fnfvima 6051 . . . . . . . . . . 11  |-  ( (
bday  Fn  No  /\  F  C_  No  /\  n  e.  F )  ->  ( bday `  n )  e.  ( bday " F
) )
108, 9mp3an1 1309 . . . . . . . . . 10  |-  ( ( F  C_  No  /\  n  e.  F )  ->  ( bday `  n )  e.  ( bday " F
) )
11103adant2 1013 . . . . . . . . 9  |-  ( ( F  C_  No  /\  F  e.  _V  /\  n  e.  F )  ->  ( bday `  n )  e.  ( bday " F
) )
12 elssuni 4192 . . . . . . . . 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 29605 . . . . . . . . 9  |-  ( bday `  n )  e.  On
15 sucelon 6551 . . . . . . . . . . 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 1016 . . . . . . . . 9  |-  ( ( F  C_  No  /\  F  e.  _V  /\  n  e.  F )  ->  U. ( bday " F )  e.  On )
18 onsssuc 4879 . . . . . . . . 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 661 . . . . . . . 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 3412 . . . . . . . . 9  |-  ( ( F  C_  No  /\  n  e.  F )  ->  n  e.  No )
22 fvnobday 29607 . . . . . . . . . 10  |-  ( n  e.  No  ->  (
n `  ( bday `  n ) )  =  (/) )
232nosgnn0i 29584 . . . . . . . . . . 11  |-  (/)  =/=  S
2423a1i 11 . . . . . . . . . 10  |-  ( n  e.  No  ->  (/)  =/=  S
)
2522, 24eqnetrd 2675 . . . . . . . . 9  |-  ( n  e.  No  ->  (
n `  ( bday `  n ) )  =/= 
S )
2621, 25syl 16 . . . . . . . 8  |-  ( ( F  C_  No  /\  n  e.  F )  ->  (
n `  ( bday `  n ) )  =/= 
S )
27263adant2 1013 . . . . . . 7  |-  ( ( F  C_  No  /\  F  e.  _V  /\  n  e.  F )  ->  (
n `  ( bday `  n ) )  =/= 
S )
28 fveq2 5774 . . . . . . . . 9  |-  ( b  =  ( bday `  n
)  ->  ( n `  b )  =  ( n `  ( bday `  n ) ) )
2928neeq1d 2659 . . . . . . . 8  |-  ( b  =  ( bday `  n
)  ->  ( (
n `  b )  =/=  S  <->  ( n `  ( bday `  n )
)  =/=  S ) )
3029rspcev 3135 . . . . . . 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 659 . . . . . 6  |-  ( ( F  C_  No  /\  F  e.  _V  /\  n  e.  F )  ->  E. b  e.  suc  U. ( bday " F ) ( n `
 b )  =/= 
S )
32313expa 1194 . . . . 5  |-  ( ( ( F  C_  No  /\  F  e.  _V )  /\  n  e.  F
)  ->  E. b  e.  suc  U. ( bday " F ) ( n `
 b )  =/= 
S )
3332ralrimiva 2796 . . . 4  |-  ( ( F  C_  No  /\  F  e.  _V )  ->  A. n  e.  F  E. b  e.  suc  U. ( bday " F ) ( n `
 b )  =/= 
S )
34 rexeq 2980 . . . . . 6  |-  ( a  =  suc  U. ( bday " F )  -> 
( E. b  e.  a  ( n `  b )  =/=  S  <->  E. b  e.  suc  U. ( bday " F ) ( n `  b
)  =/=  S ) )
3534ralbidv 2821 . . . . 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 4226 . . . 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 659 . . 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 3451 . 2  |-  ( ( F  C_  No  /\  F  e.  _V )  ->  ( bday `  ( C  X.  { S } ) ) 
C_  suc  U. ( bday " F ) )
391, 38sylan2 472 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 367    /\ w3a 971    = wceq 1399    e. wcel 1826    =/= wne 2577   A.wral 2732   E.wrex 2733   {crab 2736   _Vcvv 3034    C_ wss 3389   (/)c0 3711   {csn 3944   {cpr 3946   U.cuni 4163   |^|cint 4199   Oncon0 4792   suc csuc 4794    X. cxp 4911   "cima 4916    Fn wfn 5491   ` cfv 5496   1oc1o 7041   2oc2o 7042   Nocsur 29565   bdaycbday 29567
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-reu 2739  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-int 4200  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-1o 7048  df-2o 7049  df-no 29568  df-bday 29570
This theorem is referenced by:  nobndup  29625  nobnddown  29626
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