Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  nobndlem2 Structured version   Unicode version

Theorem nobndlem2 29670
Description: Lemma for nobndup 29677 and nobnddown 29678. 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 29669 . . . 4  |-  ( F  e.  A  ->  suc  U. ( bday " F
)  e.  On )
3 ssel2 3494 . . . . . . . . . 10  |-  ( ( F  C_  No  /\  n  e.  F )  ->  n  e.  No )
4 bdaydm 29655 . . . . . . . . . 10  |-  dom  bday  =  No
53, 4syl6eleqr 2556 . . . . . . . . 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 29653 . . . . . . . . . 10  |-  Fun  bday
8 funfvima 6148 . . . . . . . . . 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 29657 . . . . . . . 8  |-  ( bday `  n )  e.  On
14 imassrn 5358 . . . . . . . . . 10  |-  ( bday " F )  C_  ran  bday
15 bdayrn 29654 . . . . . . . . . 10  |-  ran  bday  =  On
1614, 15sseqtri 3531 . . . . . . . . 9  |-  ( bday " F )  C_  On
17 ssorduni 6620 . . . . . . . . 9  |-  ( (
bday " F )  C_  On  ->  Ord  U. ( bday " F ) )
1816, 17ax-mp 5 . . . . . . . 8  |-  Ord  U. ( bday " F )
19 ordsssuc 4973 . . . . . . . 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 29636 . . . . . . 7  |-  (/)  =/=  X
24 fvnobday 29659 . . . . . . . . 9  |-  ( n  e.  No  ->  (
n `  ( bday `  n ) )  =  (/) )
253, 24syl 16 . . . . . . . 8  |-  ( ( F  C_  No  /\  n  e.  F )  ->  (
n `  ( bday `  n ) )  =  (/) )
2625neeq1d 2734 . . . . . . 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 2734 . . . . . . 7  |-  ( b  =  ( bday `  n
)  ->  ( (
n `  b )  =/=  X  <->  ( n `  ( bday `  n )
)  =/=  X ) )
3029rspcev 3210 . . . . . 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 2871 . . . 4  |-  ( F 
C_  No  ->  A. n  e.  F  E. b  e.  suc  U. ( bday " F ) ( n `
 b )  =/= 
X )
33 rexeq 3055 . . . . . 6  |-  ( a  =  suc  U. ( bday " F )  -> 
( E. b  e.  a  ( n `  b )  =/=  X  <->  E. b  e.  suc  U. ( bday " F ) ( n `  b
)  =/=  X ) )
3433ralbidv 2896 . . . . 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 3210 . . . 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 6636 . . 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 2549 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 1395    e. wcel 1819    =/= wne 2652   A.wral 2807   E.wrex 2808   {crab 2811    C_ wss 3471   (/)c0 3793   {cpr 4034   U.cuni 4251   |^|cint 4288   Ord word 4886   Oncon0 4887   suc csuc 4889   dom cdm 5008   ran crn 5009   "cima 5011   Fun wfun 5588   ` cfv 5594   1oc1o 7141   2oc2o 7142   Nocsur 29617   bdaycbday 29619
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  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 7148  df-2o 7149  df-no 29620  df-bday 29622
This theorem is referenced by:  nobndlem3  29671  nobndup  29677  nobnddown  29678
  Copyright terms: Public domain W3C validator