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Theorem nobndlem2 27856
Description: Lemma for nobndup 27863 and nobnddown 27864. 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 27855 . . . 4  |-  ( F  e.  A  ->  suc  U. ( bday " F
)  e.  On )
3 ssel2 3372 . . . . . . . . . 10  |-  ( ( F  C_  No  /\  n  e.  F )  ->  n  e.  No )
4 bdaydm 27841 . . . . . . . . . 10  |-  dom  bday  =  No
53, 4syl6eleqr 2534 . . . . . . . . 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 27839 . . . . . . . . . 10  |-  Fun  bday
8 funfvima 5973 . . . . . . . . . 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 4142 . . . . . . . 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 27843 . . . . . . . 8  |-  ( bday `  n )  e.  On
14 imassrn 5201 . . . . . . . . . 10  |-  ( bday " F )  C_  ran  bday
15 bdayrn 27840 . . . . . . . . . 10  |-  ran  bday  =  On
1614, 15sseqtri 3409 . . . . . . . . 9  |-  ( bday " F )  C_  On
17 ssorduni 6418 . . . . . . . . 9  |-  ( (
bday " F )  C_  On  ->  Ord  U. ( bday " F ) )
1816, 17ax-mp 5 . . . . . . . 8  |-  Ord  U. ( bday " F )
19 ordsssuc 4826 . . . . . . . 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 27822 . . . . . . 7  |-  (/)  =/=  X
24 fvnobday 27845 . . . . . . . . 9  |-  ( n  e.  No  ->  (
n `  ( bday `  n ) )  =  (/) )
253, 24syl 16 . . . . . . . 8  |-  ( ( F  C_  No  /\  n  e.  F )  ->  (
n `  ( bday `  n ) )  =  (/) )
2625neeq1d 2641 . . . . . . 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 5712 . . . . . . . 8  |-  ( b  =  ( bday `  n
)  ->  ( n `  b )  =  ( n `  ( bday `  n ) ) )
2928neeq1d 2641 . . . . . . 7  |-  ( b  =  ( bday `  n
)  ->  ( (
n `  b )  =/=  X  <->  ( n `  ( bday `  n )
)  =/=  X ) )
3029rspcev 3094 . . . . . 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 2820 . . . 4  |-  ( F 
C_  No  ->  A. n  e.  F  E. b  e.  suc  U. ( bday " F ) ( n `
 b )  =/= 
X )
33 rexeq 2939 . . . . . 6  |-  ( a  =  suc  U. ( bday " F )  -> 
( E. b  e.  a  ( n `  b )  =/=  X  <->  E. b  e.  suc  U. ( bday " F ) ( n `  b
)  =/=  X ) )
3433ralbidv 2756 . . . . 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 3094 . . . 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 6434 . . 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 2527 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 1369    e. wcel 1756    =/= wne 2620   A.wral 2736   E.wrex 2737   {crab 2740    C_ wss 3349   (/)c0 3658   {cpr 3900   U.cuni 4112   |^|cint 4149   Ord word 4739   Oncon0 4740   suc csuc 4742   dom cdm 4861   ran crn 4862   "cima 4864   Fun wfun 5433   ` cfv 5439   1oc1o 6934   2oc2o 6935   Nocsur 27803   bdaycbday 27805
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-int 4150  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-on 4744  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-1o 6941  df-2o 6942  df-no 27806  df-bday 27808
This theorem is referenced by:  nobndlem3  27857  nobndup  27863  nobnddown  27864
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