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Theorem nobndlem2 29380
Description: Lemma for nobndup 29387 and nobnddown 29388. 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 29379 . . . 4  |-  ( F  e.  A  ->  suc  U. ( bday " F
)  e.  On )
3 ssel2 3504 . . . . . . . . . 10  |-  ( ( F  C_  No  /\  n  e.  F )  ->  n  e.  No )
4 bdaydm 29365 . . . . . . . . . 10  |-  dom  bday  =  No
53, 4syl6eleqr 2566 . . . . . . . . 9  |-  ( ( F  C_  No  /\  n  e.  F )  ->  n  e.  dom  bday )
6 simpr 461 . . . . . . . . 9  |-  ( ( F  C_  No  /\  n  e.  F )  ->  n  e.  F )
7 bdayfun 29363 . . . . . . . . . 10  |-  Fun  bday
8 funfvima 6146 . . . . . . . . . 10  |-  ( ( Fun  bday  /\  n  e.  dom  bday )  ->  (
n  e.  F  -> 
( bday `  n )  e.  ( bday " F
) ) )
97, 8mpan 670 . . . . . . . . 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 4281 . . . . . . . 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 29367 . . . . . . . 8  |-  ( bday `  n )  e.  On
14 imassrn 5354 . . . . . . . . . 10  |-  ( bday " F )  C_  ran  bday
15 bdayrn 29364 . . . . . . . . . 10  |-  ran  bday  =  On
1614, 15sseqtri 3541 . . . . . . . . 9  |-  ( bday " F )  C_  On
17 ssorduni 6616 . . . . . . . . 9  |-  ( (
bday " F )  C_  On  ->  Ord  U. ( bday " F ) )
1816, 17ax-mp 5 . . . . . . . 8  |-  Ord  U. ( bday " F )
19 ordsssuc 4970 . . . . . . . 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 672 . . . . . . 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 29346 . . . . . . 7  |-  (/)  =/=  X
24 fvnobday 29369 . . . . . . . . 9  |-  ( n  e.  No  ->  (
n `  ( bday `  n ) )  =  (/) )
253, 24syl 16 . . . . . . . 8  |-  ( ( F  C_  No  /\  n  e.  F )  ->  (
n `  ( bday `  n ) )  =  (/) )
2625neeq1d 2744 . . . . . . 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 5872 . . . . . . . 8  |-  ( b  =  ( bday `  n
)  ->  ( n `  b )  =  ( n `  ( bday `  n ) ) )
2928neeq1d 2744 . . . . . . 7  |-  ( b  =  ( bday `  n
)  ->  ( (
n `  b )  =/=  X  <->  ( n `  ( bday `  n )
)  =/=  X ) )
3029rspcev 3219 . . . . . 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 661 . . . . 5  |-  ( ( F  C_  No  /\  n  e.  F )  ->  E. b  e.  suc  U. ( bday " F ) ( n `
 b )  =/= 
X )
3231ralrimiva 2881 . . . 4  |-  ( F 
C_  No  ->  A. n  e.  F  E. b  e.  suc  U. ( bday " F ) ( n `
 b )  =/= 
X )
33 rexeq 3064 . . . . . 6  |-  ( a  =  suc  U. ( bday " F )  -> 
( E. b  e.  a  ( n `  b )  =/=  X  <->  E. b  e.  suc  U. ( bday " F ) ( n `  b
)  =/=  X ) )
3433ralbidv 2906 . . . . 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 3219 . . . 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 478 . . 3  |-  ( ( F  C_  No  /\  F  e.  A )  ->  E. a  e.  On  A. n  e.  F  E. b  e.  a  ( n `  b )  =/=  X
)
37 onintrab2 6632 . . 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 2559 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 1379    e. wcel 1767    =/= wne 2662   A.wral 2817   E.wrex 2818   {crab 2821    C_ wss 3481   (/)c0 3790   {cpr 4035   U.cuni 4251   |^|cint 4288   Ord word 4883   Oncon0 4884   suc csuc 4886   dom cdm 5005   ran crn 5006   "cima 5008   Fun wfun 5588   ` cfv 5594   1oc1o 7135   2oc2o 7136   Nocsur 29327   bdaycbday 29329
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-1o 7142  df-2o 7143  df-no 29330  df-bday 29332
This theorem is referenced by:  nobndlem3  29381  nobndup  29387  nobnddown  29388
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