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Theorem nobndlem2 25561
Description: Lemma for nobndup 25568 and nobnddown 25569. 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 25560 . . . 4  |-  ( F  e.  A  ->  suc  U. ( bday " F
)  e.  On )
3 ssel2 3303 . . . . . . . . . 10  |-  ( ( F  C_  No  /\  n  e.  F )  ->  n  e.  No )
4 bdaydm 25546 . . . . . . . . . 10  |-  dom  bday  =  No
53, 4syl6eleqr 2495 . . . . . . . . 9  |-  ( ( F  C_  No  /\  n  e.  F )  ->  n  e.  dom  bday )
6 simpr 448 . . . . . . . . 9  |-  ( ( F  C_  No  /\  n  e.  F )  ->  n  e.  F )
7 bdayfun 25544 . . . . . . . . . 10  |-  Fun  bday
8 funfvima 5932 . . . . . . . . . 10  |-  ( ( Fun  bday  /\  n  e.  dom  bday )  ->  (
n  e.  F  -> 
( bday `  n )  e.  ( bday " F
) ) )
97, 8mpan 652 . . . . . . . . 9  |-  ( n  e.  dom  bday  ->  ( n  e.  F  -> 
( bday `  n )  e.  ( bday " F
) ) )
105, 6, 9sylc 58 . . . . . . . 8  |-  ( ( F  C_  No  /\  n  e.  F )  ->  ( bday `  n )  e.  ( bday " F
) )
11 elssuni 4003 . . . . . . . 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 25548 . . . . . . . 8  |-  ( bday `  n )  e.  On
14 imassrn 5175 . . . . . . . . . 10  |-  ( bday " F )  C_  ran  bday
15 bdayrn 25545 . . . . . . . . . 10  |-  ran  bday  =  On
1614, 15sseqtri 3340 . . . . . . . . 9  |-  ( bday " F )  C_  On
17 ssorduni 4725 . . . . . . . . 9  |-  ( (
bday " F )  C_  On  ->  Ord  U. ( bday " F ) )
1816, 17ax-mp 8 . . . . . . . 8  |-  Ord  U. ( bday " F )
19 ordsssuc 4627 . . . . . . . 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 654 . . . . . . 7  |-  ( (
bday `  n )  C_ 
U. ( bday " F
)  <->  ( bday `  n
)  e.  suc  U. ( bday " F ) )
2112, 20sylib 189 . . . . . 6  |-  ( ( F  C_  No  /\  n  e.  F )  ->  ( bday `  n )  e. 
suc  U. ( bday " F
) )
22 nobndlem2.1 . . . . . . . 8  |-  X  e. 
{ 1o ,  2o }
2322nosgnn0i 25527 . . . . . . 7  |-  (/)  =/=  X
24 fvnobday 25550 . . . . . . . . 9  |-  ( n  e.  No  ->  (
n `  ( bday `  n ) )  =  (/) )
253, 24syl 16 . . . . . . . 8  |-  ( ( F  C_  No  /\  n  e.  F )  ->  (
n `  ( bday `  n ) )  =  (/) )
2625neeq1d 2580 . . . . . . 7  |-  ( ( F  C_  No  /\  n  e.  F )  ->  (
( n `  ( bday `  n ) )  =/=  X  <->  (/)  =/=  X
) )
2723, 26mpbiri 225 . . . . . 6  |-  ( ( F  C_  No  /\  n  e.  F )  ->  (
n `  ( bday `  n ) )  =/= 
X )
28 fveq2 5687 . . . . . . . 8  |-  ( b  =  ( bday `  n
)  ->  ( n `  b )  =  ( n `  ( bday `  n ) ) )
2928neeq1d 2580 . . . . . . 7  |-  ( b  =  ( bday `  n
)  ->  ( (
n `  b )  =/=  X  <->  ( n `  ( bday `  n )
)  =/=  X ) )
3029rspcev 3012 . . . . . 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 643 . . . . 5  |-  ( ( F  C_  No  /\  n  e.  F )  ->  E. b  e.  suc  U. ( bday " F ) ( n `
 b )  =/= 
X )
3231ralrimiva 2749 . . . 4  |-  ( F 
C_  No  ->  A. n  e.  F  E. b  e.  suc  U. ( bday " F ) ( n `
 b )  =/= 
X )
33 rexeq 2865 . . . . . 6  |-  ( a  =  suc  U. ( bday " F )  -> 
( E. b  e.  a  ( n `  b )  =/=  X  <->  E. b  e.  suc  U. ( bday " F ) ( n `  b
)  =/=  X ) )
3433ralbidv 2686 . . . . 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 3012 . . . 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 465 . . 3  |-  ( ( F  C_  No  /\  F  e.  A )  ->  E. a  e.  On  A. n  e.  F  E. b  e.  a  ( n `  b )  =/=  X
)
37 onintrab2 4741 . . 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 189 . 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 2488 1  |-  ( ( F  C_  No  /\  F  e.  A )  ->  C  e.  On )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2567   A.wral 2666   E.wrex 2667   {crab 2670    C_ wss 3280   (/)c0 3588   {cpr 3775   U.cuni 3975   |^|cint 4010   Ord word 4540   Oncon0 4541   suc csuc 4543   dom cdm 4837   ran crn 4838   "cima 4840   Fun wfun 5407   ` cfv 5413   1oc1o 6676   2oc2o 6677   Nocsur 25508   bdaycbday 25510
This theorem is referenced by:  nobndlem3  25562  nobndup  25568  nobnddown  25569
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-suc 4547  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-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-1o 6683  df-2o 6684  df-no 25511  df-bday 25513
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