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Theorem nobndlem2 27763
Description: Lemma for nobndup 27770 and nobnddown 27771. Show a particular abstraction is an ordinal. (Contributed by Scott Fenton, 17-Aug-2011.)
Hypotheses
Ref Expression
nobndlem2.1  |-  X  e. 
{ 1o ,  2o }
nobndlem2.2  |-  C  = 
|^| { a  e.  On  |  A. n  e.  F  E. b  e.  a 
( n `  b
)  =/=  X }
Assertion
Ref Expression
nobndlem2  |-  ( ( F  C_  No  /\  F  e.  A )  ->  C  e.  On )
Distinct variable groups:    F, a,
b, n    X, a,
b
Allowed substitution hints:    A( n, a, b)    C( n, a, b)    X( n)

Proof of Theorem nobndlem2
StepHypRef Expression
1 nobndlem2.2 . 2  |-  C  = 
|^| { a  e.  On  |  A. n  e.  F  E. b  e.  a 
( n `  b
)  =/=  X }
2 nobndlem1 27762 . . . 4  |-  ( F  e.  A  ->  suc  U. ( bday " F
)  e.  On )
3 ssel2 3348 . . . . . . . . . 10  |-  ( ( F  C_  No  /\  n  e.  F )  ->  n  e.  No )
4 bdaydm 27748 . . . . . . . . . 10  |-  dom  bday  =  No
53, 4syl6eleqr 2532 . . . . . . . . 9  |-  ( ( F  C_  No  /\  n  e.  F )  ->  n  e.  dom  bday )
6 simpr 458 . . . . . . . . 9  |-  ( ( F  C_  No  /\  n  e.  F )  ->  n  e.  F )
7 bdayfun 27746 . . . . . . . . . 10  |-  Fun  bday
8 funfvima 5949 . . . . . . . . . 10  |-  ( ( Fun  bday  /\  n  e.  dom  bday )  ->  (
n  e.  F  -> 
( bday `  n )  e.  ( bday " F
) ) )
97, 8mpan 665 . . . . . . . . 9  |-  ( n  e.  dom  bday  ->  ( n  e.  F  -> 
( bday `  n )  e.  ( bday " F
) ) )
105, 6, 9sylc 60 . . . . . . . 8  |-  ( ( F  C_  No  /\  n  e.  F )  ->  ( bday `  n )  e.  ( bday " F
) )
11 elssuni 4118 . . . . . . . 8  |-  ( (
bday `  n )  e.  ( bday " F
)  ->  ( bday `  n )  C_  U. ( bday " F ) )
1210, 11syl 16 . . . . . . 7  |-  ( ( F  C_  No  /\  n  e.  F )  ->  ( bday `  n )  C_  U. ( bday " F
) )
13 bdayelon 27750 . . . . . . . 8  |-  ( bday `  n )  e.  On
14 imassrn 5177 . . . . . . . . . 10  |-  ( bday " F )  C_  ran  bday
15 bdayrn 27747 . . . . . . . . . 10  |-  ran  bday  =  On
1614, 15sseqtri 3385 . . . . . . . . 9  |-  ( bday " F )  C_  On
17 ssorduni 6396 . . . . . . . . 9  |-  ( (
bday " F )  C_  On  ->  Ord  U. ( bday " F ) )
1816, 17ax-mp 5 . . . . . . . 8  |-  Ord  U. ( bday " F )
19 ordsssuc 4801 . . . . . . . 8  |-  ( ( ( bday `  n
)  e.  On  /\  Ord  U. ( bday " F
) )  ->  (
( bday `  n )  C_ 
U. ( bday " F
)  <->  ( bday `  n
)  e.  suc  U. ( bday " F ) ) )
2013, 18, 19mp2an 667 . . . . . . 7  |-  ( (
bday `  n )  C_ 
U. ( bday " F
)  <->  ( bday `  n
)  e.  suc  U. ( bday " F ) )
2112, 20sylib 196 . . . . . 6  |-  ( ( F  C_  No  /\  n  e.  F )  ->  ( bday `  n )  e. 
suc  U. ( bday " F
) )
22 nobndlem2.1 . . . . . . . 8  |-  X  e. 
{ 1o ,  2o }
2322nosgnn0i 27729 . . . . . . 7  |-  (/)  =/=  X
24 fvnobday 27752 . . . . . . . . 9  |-  ( n  e.  No  ->  (
n `  ( bday `  n ) )  =  (/) )
253, 24syl 16 . . . . . . . 8  |-  ( ( F  C_  No  /\  n  e.  F )  ->  (
n `  ( bday `  n ) )  =  (/) )
2625neeq1d 2619 . . . . . . 7  |-  ( ( F  C_  No  /\  n  e.  F )  ->  (
( n `  ( bday `  n ) )  =/=  X  <->  (/)  =/=  X
) )
2723, 26mpbiri 233 . . . . . 6  |-  ( ( F  C_  No  /\  n  e.  F )  ->  (
n `  ( bday `  n ) )  =/= 
X )
28 fveq2 5688 . . . . . . . 8  |-  ( b  =  ( bday `  n
)  ->  ( n `  b )  =  ( n `  ( bday `  n ) ) )
2928neeq1d 2619 . . . . . . 7  |-  ( b  =  ( bday `  n
)  ->  ( (
n `  b )  =/=  X  <->  ( n `  ( bday `  n )
)  =/=  X ) )
3029rspcev 3070 . . . . . 6  |-  ( ( ( bday `  n
)  e.  suc  U. ( bday " F )  /\  ( n `  ( bday `  n )
)  =/=  X )  ->  E. b  e.  suc  U. ( bday " F
) ( n `  b )  =/=  X
)
3121, 27, 30syl2anc 656 . . . . 5  |-  ( ( F  C_  No  /\  n  e.  F )  ->  E. b  e.  suc  U. ( bday " F ) ( n `
 b )  =/= 
X )
3231ralrimiva 2797 . . . 4  |-  ( F 
C_  No  ->  A. n  e.  F  E. b  e.  suc  U. ( bday " F ) ( n `
 b )  =/= 
X )
33 rexeq 2916 . . . . . 6  |-  ( a  =  suc  U. ( bday " F )  -> 
( E. b  e.  a  ( n `  b )  =/=  X  <->  E. b  e.  suc  U. ( bday " F ) ( n `  b
)  =/=  X ) )
3433ralbidv 2733 . . . . 5  |-  ( a  =  suc  U. ( bday " F )  -> 
( A. n  e.  F  E. b  e.  a  ( n `  b )  =/=  X  <->  A. n  e.  F  E. b  e.  suc  U. ( bday " F ) ( n `  b )  =/=  X ) )
3534rspcev 3070 . . . 4  |-  ( ( suc  U. ( bday " F )  e.  On  /\ 
A. n  e.  F  E. b  e.  suc  U. ( bday " F
) ( n `  b )  =/=  X
)  ->  E. a  e.  On  A. n  e.  F  E. b  e.  a  ( n `  b )  =/=  X
)
362, 32, 35syl2anr 475 . . 3  |-  ( ( F  C_  No  /\  F  e.  A )  ->  E. a  e.  On  A. n  e.  F  E. b  e.  a  ( n `  b )  =/=  X
)
37 onintrab2 6412 . . 3  |-  ( E. a  e.  On  A. n  e.  F  E. b  e.  a  (
n `  b )  =/=  X  <->  |^| { a  e.  On  |  A. n  e.  F  E. b  e.  a  ( n `  b )  =/=  X }  e.  On )
3836, 37sylib 196 . 2  |-  ( ( F  C_  No  /\  F  e.  A )  ->  |^| { a  e.  On  |  A. n  e.  F  E. b  e.  a  (
n `  b )  =/=  X }  e.  On )
391, 38syl5eqel 2525 1  |-  ( ( F  C_  No  /\  F  e.  A )  ->  C  e.  On )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1364    e. wcel 1761    =/= wne 2604   A.wral 2713   E.wrex 2714   {crab 2717    C_ wss 3325   (/)c0 3634   {cpr 3876   U.cuni 4088   |^|cint 4125   Ord word 4714   Oncon0 4715   suc csuc 4717   dom cdm 4836   ran crn 4837   "cima 4839   Fun wfun 5409   ` cfv 5415   1oc1o 6909   2oc2o 6910   Nocsur 27710   bdaycbday 27712
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-1o 6916  df-2o 6917  df-no 27713  df-bday 27715
This theorem is referenced by:  nobndlem3  27764  nobndup  27770  nobnddown  27771
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