Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  nodenselem4 Structured version   Unicode version

Theorem nodenselem4 25631
Description: Lemma for nodense 25636. Show that a particular abstraction is an ordinal. (Contributed by Scott Fenton, 16-Jun-2011.)
Assertion
Ref Expression
nodenselem4  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  A < s B )  ->  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  On )
Distinct variable groups:    A, a    B, a

Proof of Theorem nodenselem4
StepHypRef Expression
1 ssrab2 3420 . 2  |-  { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  C_  On
2 sltirr 25617 . . . . . . 7  |-  ( A  e.  No  ->  -.  A < s A )
3 breq2 4208 . . . . . . . . 9  |-  ( A  =  B  ->  ( A < s A  <->  A < s B ) )
43biimprcd 217 . . . . . . . 8  |-  ( A < s B  -> 
( A  =  B  ->  A < s A ) )
54con3d 127 . . . . . . 7  |-  ( A < s B  -> 
( -.  A <
s A  ->  -.  A  =  B )
)
62, 5syl5com 28 . . . . . 6  |-  ( A  e.  No  ->  ( A < s B  ->  -.  A  =  B
) )
76adantr 452 . . . . 5  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( A < s B  ->  -.  A  =  B ) )
8 nofnbday 25599 . . . . . . . 8  |-  ( A  e.  No  ->  A  Fn  ( bday `  A
) )
9 nofnbday 25599 . . . . . . . 8  |-  ( B  e.  No  ->  B  Fn  ( bday `  B
) )
10 eqfnfv2 5820 . . . . . . . 8  |-  ( ( A  Fn  ( bday `  A )  /\  B  Fn  ( bday `  B
) )  ->  ( A  =  B  <->  ( ( bday `  A )  =  ( bday `  B
)  /\  A. a  e.  ( bday `  A
) ( A `  a )  =  ( B `  a ) ) ) )
118, 9, 10syl2an 464 . . . . . . 7  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( A  =  B  <-> 
( ( bday `  A
)  =  ( bday `  B )  /\  A. a  e.  ( bday `  A ) ( A `
 a )  =  ( B `  a
) ) ) )
1211notbid 286 . . . . . 6  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( -.  A  =  B  <->  -.  ( ( bday `  A )  =  ( bday `  B
)  /\  A. a  e.  ( bday `  A
) ( A `  a )  =  ( B `  a ) ) ) )
13 ianor 475 . . . . . . 7  |-  ( -.  ( ( bday `  A
)  =  ( bday `  B )  /\  A. a  e.  ( bday `  A ) ( A `
 a )  =  ( B `  a
) )  <->  ( -.  ( bday `  A )  =  ( bday `  B
)  \/  -.  A. a  e.  ( bday `  A ) ( A `
 a )  =  ( B `  a
) ) )
14 bdayelon 25627 . . . . . . . . . . . 12  |-  ( bday `  A )  e.  On
1514onordi 4678 . . . . . . . . . . 11  |-  Ord  ( bday `  A )
16 bdayelon 25627 . . . . . . . . . . . 12  |-  ( bday `  B )  e.  On
1716onordi 4678 . . . . . . . . . . 11  |-  Ord  ( bday `  B )
18 ordtri3 4609 . . . . . . . . . . 11  |-  ( ( Ord  ( bday `  A
)  /\  Ord  ( bday `  B ) )  -> 
( ( bday `  A
)  =  ( bday `  B )  <->  -.  (
( bday `  A )  e.  ( bday `  B
)  \/  ( bday `  B )  e.  (
bday `  A )
) ) )
1915, 17, 18mp2an 654 . . . . . . . . . 10  |-  ( (
bday `  A )  =  ( bday `  B
)  <->  -.  ( ( bday `  A )  e.  ( bday `  B
)  \/  ( bday `  B )  e.  (
bday `  A )
) )
2019con2bii 323 . . . . . . . . 9  |-  ( ( ( bday `  A
)  e.  ( bday `  B )  \/  ( bday `  B )  e.  ( bday `  A
) )  <->  -.  ( bday `  A )  =  ( bday `  B
) )
21 nodenselem3 25630 . . . . . . . . . 10  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( ( bday `  A
)  e.  ( bday `  B )  ->  E. a  e.  On  ( A `  a )  =/=  ( B `  a )
) )
22 nodenselem3 25630 . . . . . . . . . . . 12  |-  ( ( B  e.  No  /\  A  e.  No )  ->  ( ( bday `  B
)  e.  ( bday `  A )  ->  E. a  e.  On  ( B `  a )  =/=  ( A `  a )
) )
23 necom 2679 . . . . . . . . . . . . 13  |-  ( ( B `  a )  =/=  ( A `  a )  <->  ( A `  a )  =/=  ( B `  a )
)
2423rexbii 2722 . . . . . . . . . . . 12  |-  ( E. a  e.  On  ( B `  a )  =/=  ( A `  a
)  <->  E. a  e.  On  ( A `  a )  =/=  ( B `  a ) )
2522, 24syl6ib 218 . . . . . . . . . . 11  |-  ( ( B  e.  No  /\  A  e.  No )  ->  ( ( bday `  B
)  e.  ( bday `  A )  ->  E. a  e.  On  ( A `  a )  =/=  ( B `  a )
) )
2625ancoms 440 . . . . . . . . . 10  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( ( bday `  B
)  e.  ( bday `  A )  ->  E. a  e.  On  ( A `  a )  =/=  ( B `  a )
) )
2721, 26jaod 370 . . . . . . . . 9  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( ( ( bday `  A )  e.  (
bday `  B )  \/  ( bday `  B
)  e.  ( bday `  A ) )  ->  E. a  e.  On  ( A `  a )  =/=  ( B `  a ) ) )
2820, 27syl5bir 210 . . . . . . . 8  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( -.  ( bday `  A )  =  (
bday `  B )  ->  E. a  e.  On  ( A `  a )  =/=  ( B `  a ) ) )
29 rexnal 2708 . . . . . . . . . 10  |-  ( E. a  e.  ( bday `  A )  -.  ( A `  a )  =  ( B `  a )  <->  -.  A. a  e.  ( bday `  A
) ( A `  a )  =  ( B `  a ) )
3014onssi 4809 . . . . . . . . . . . 12  |-  ( bday `  A )  C_  On
31 ssrexv 3400 . . . . . . . . . . . 12  |-  ( (
bday `  A )  C_  On  ->  ( E. a  e.  ( bday `  A )  -.  ( A `  a )  =  ( B `  a )  ->  E. a  e.  On  -.  ( A `
 a )  =  ( B `  a
) ) )
3230, 31ax-mp 8 . . . . . . . . . . 11  |-  ( E. a  e.  ( bday `  A )  -.  ( A `  a )  =  ( B `  a )  ->  E. a  e.  On  -.  ( A `
 a )  =  ( B `  a
) )
33 df-ne 2600 . . . . . . . . . . . 12  |-  ( ( A `  a )  =/=  ( B `  a )  <->  -.  ( A `  a )  =  ( B `  a ) )
3433rexbii 2722 . . . . . . . . . . 11  |-  ( E. a  e.  On  ( A `  a )  =/=  ( B `  a
)  <->  E. a  e.  On  -.  ( A `  a
)  =  ( B `
 a ) )
3532, 34sylibr 204 . . . . . . . . . 10  |-  ( E. a  e.  ( bday `  A )  -.  ( A `  a )  =  ( B `  a )  ->  E. a  e.  On  ( A `  a )  =/=  ( B `  a )
)
3629, 35sylbir 205 . . . . . . . . 9  |-  ( -. 
A. a  e.  (
bday `  A )
( A `  a
)  =  ( B `
 a )  ->  E. a  e.  On  ( A `  a )  =/=  ( B `  a ) )
3736a1i 11 . . . . . . . 8  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( -.  A. a  e.  ( bday `  A
) ( A `  a )  =  ( B `  a )  ->  E. a  e.  On  ( A `  a )  =/=  ( B `  a ) ) )
3828, 37jaod 370 . . . . . . 7  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( ( -.  ( bday `  A )  =  ( bday `  B
)  \/  -.  A. a  e.  ( bday `  A ) ( A `
 a )  =  ( B `  a
) )  ->  E. a  e.  On  ( A `  a )  =/=  ( B `  a )
) )
3913, 38syl5bi 209 . . . . . 6  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( -.  ( (
bday `  A )  =  ( bday `  B
)  /\  A. a  e.  ( bday `  A
) ( A `  a )  =  ( B `  a ) )  ->  E. a  e.  On  ( A `  a )  =/=  ( B `  a )
) )
4012, 39sylbid 207 . . . . 5  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( -.  A  =  B  ->  E. a  e.  On  ( A `  a )  =/=  ( B `  a )
) )
417, 40syld 42 . . . 4  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( A < s B  ->  E. a  e.  On  ( A `  a )  =/=  ( B `  a ) ) )
4241imp 419 . . 3  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  A < s B )  ->  E. a  e.  On  ( A `  a )  =/=  ( B `  a )
)
43 rabn0 3639 . . 3  |-  ( { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) }  =/=  (/)  <->  E. a  e.  On  ( A `  a )  =/=  ( B `  a ) )
4442, 43sylibr 204 . 2  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  A < s B )  ->  { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  =/=  (/) )
45 oninton 4772 . 2  |-  ( ( { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  C_  On  /\  {
a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) }  =/=  (/) )  ->  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  On )
461, 44, 45sylancr 645 1  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  A < s B )  ->  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  On )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   E.wrex 2698   {crab 2701    C_ wss 3312   (/)c0 3620   |^|cint 4042   class class class wbr 4204   Ord word 4572   Oncon0 4573    Fn wfn 5441   ` cfv 5446   Nocsur 25587   < scslt 25588   bdaycbday 25589
This theorem is referenced by:  nodenselem5  25632  nodenselem6  25633  nodenselem7  25634  nodense  25636
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-suc 4579  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-1o 6716  df-2o 6717  df-no 25590  df-slt 25591  df-bday 25592
  Copyright terms: Public domain W3C validator