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Theorem nodenselem4 30522
Description: Lemma for nodense 30527. 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 3489 . 2  |-  { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  C_  On
2 sltirr 30508 . . . . . . 7  |-  ( A  e.  No  ->  -.  A <s A )
3 breq2 4370 . . . . . . . . 9  |-  ( A  =  B  ->  ( A <s A  <->  A <s B ) )
43biimprcd 228 . . . . . . . 8  |-  ( A <s B  -> 
( A  =  B  ->  A <s
A ) )
54con3d 138 . . . . . . 7  |-  ( A <s B  -> 
( -.  A <s A  ->  -.  A  =  B ) )
62, 5syl5com 31 . . . . . 6  |-  ( A  e.  No  ->  ( A <s B  ->  -.  A  =  B
) )
76adantr 466 . . . . 5  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( A <s
B  ->  -.  A  =  B ) )
8 nofnbday 30490 . . . . . . . 8  |-  ( A  e.  No  ->  A  Fn  ( bday `  A
) )
9 nofnbday 30490 . . . . . . . 8  |-  ( B  e.  No  ->  B  Fn  ( bday `  B
) )
10 eqfnfv2 5936 . . . . . . . 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 479 . . . . . . 7  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( A  =  B  <-> 
( ( bday `  A
)  =  ( bday `  B )  /\  A. a  e.  ( bday `  A ) ( A `
 a )  =  ( B `  a
) ) ) )
1211notbid 295 . . . . . 6  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( -.  A  =  B  <->  -.  ( ( bday `  A )  =  ( bday `  B
)  /\  A. a  e.  ( bday `  A
) ( A `  a )  =  ( B `  a ) ) ) )
13 ianor 490 . . . . . . 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 30518 . . . . . . . . . . . 12  |-  ( bday `  A )  e.  On
1514onordi 5489 . . . . . . . . . . 11  |-  Ord  ( bday `  A )
16 bdayelon 30518 . . . . . . . . . . . 12  |-  ( bday `  B )  e.  On
1716onordi 5489 . . . . . . . . . . 11  |-  Ord  ( bday `  B )
18 ordtri3 5421 . . . . . . . . . . 11  |-  ( ( Ord  ( bday `  A
)  /\  Ord  ( bday `  B ) )  -> 
( ( bday `  A
)  =  ( bday `  B )  <->  -.  (
( bday `  A )  e.  ( bday `  B
)  \/  ( bday `  B )  e.  (
bday `  A )
) ) )
1915, 17, 18mp2an 676 . . . . . . . . . 10  |-  ( (
bday `  A )  =  ( bday `  B
)  <->  -.  ( ( bday `  A )  e.  ( bday `  B
)  \/  ( bday `  B )  e.  (
bday `  A )
) )
2019con2bii 333 . . . . . . . . 9  |-  ( ( ( bday `  A
)  e.  ( bday `  B )  \/  ( bday `  B )  e.  ( bday `  A
) )  <->  -.  ( bday `  A )  =  ( bday `  B
) )
21 nodenselem3 30521 . . . . . . . . . 10  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( ( bday `  A
)  e.  ( bday `  B )  ->  E. a  e.  On  ( A `  a )  =/=  ( B `  a )
) )
22 nodenselem3 30521 . . . . . . . . . . . 12  |-  ( ( B  e.  No  /\  A  e.  No )  ->  ( ( bday `  B
)  e.  ( bday `  A )  ->  E. a  e.  On  ( B `  a )  =/=  ( A `  a )
) )
23 necom 2654 . . . . . . . . . . . . 13  |-  ( ( B `  a )  =/=  ( A `  a )  <->  ( A `  a )  =/=  ( B `  a )
)
2423rexbii 2866 . . . . . . . . . . . 12  |-  ( E. a  e.  On  ( B `  a )  =/=  ( A `  a
)  <->  E. a  e.  On  ( A `  a )  =/=  ( B `  a ) )
2522, 24syl6ib 229 . . . . . . . . . . 11  |-  ( ( B  e.  No  /\  A  e.  No )  ->  ( ( bday `  B
)  e.  ( bday `  A )  ->  E. a  e.  On  ( A `  a )  =/=  ( B `  a )
) )
2625ancoms 454 . . . . . . . . . 10  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( ( bday `  B
)  e.  ( bday `  A )  ->  E. a  e.  On  ( A `  a )  =/=  ( B `  a )
) )
2721, 26jaod 381 . . . . . . . . 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 221 . . . . . . . 8  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( -.  ( bday `  A )  =  (
bday `  B )  ->  E. a  e.  On  ( A `  a )  =/=  ( B `  a ) ) )
29 rexnal 2813 . . . . . . . . . 10  |-  ( E. a  e.  ( bday `  A )  -.  ( A `  a )  =  ( B `  a )  <->  -.  A. a  e.  ( bday `  A
) ( A `  a )  =  ( B `  a ) )
3014onssi 6622 . . . . . . . . . . . 12  |-  ( bday `  A )  C_  On
31 ssrexv 3469 . . . . . . . . . . . 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 5 . . . . . . . . . . 11  |-  ( E. a  e.  ( bday `  A )  -.  ( A `  a )  =  ( B `  a )  ->  E. a  e.  On  -.  ( A `
 a )  =  ( B `  a
) )
33 df-ne 2601 . . . . . . . . . . . 12  |-  ( ( A `  a )  =/=  ( B `  a )  <->  -.  ( A `  a )  =  ( B `  a ) )
3433rexbii 2866 . . . . . . . . . . 11  |-  ( E. a  e.  On  ( A `  a )  =/=  ( B `  a
)  <->  E. a  e.  On  -.  ( A `  a
)  =  ( B `
 a ) )
3532, 34sylibr 215 . . . . . . . . . 10  |-  ( E. a  e.  ( bday `  A )  -.  ( A `  a )  =  ( B `  a )  ->  E. a  e.  On  ( A `  a )  =/=  ( B `  a )
)
3629, 35sylbir 216 . . . . . . . . 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 381 . . . . . . 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 220 . . . . . 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 218 . . . . 5  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( -.  A  =  B  ->  E. a  e.  On  ( A `  a )  =/=  ( B `  a )
) )
417, 40syld 45 . . . 4  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( A <s
B  ->  E. a  e.  On  ( A `  a )  =/=  ( B `  a )
) )
4241imp 430 . . 3  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  A <s
B )  ->  E. a  e.  On  ( A `  a )  =/=  ( B `  a )
)
43 rabn0 3725 . . 3  |-  ( { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) }  =/=  (/)  <->  E. a  e.  On  ( A `  a )  =/=  ( B `  a ) )
4442, 43sylibr 215 . 2  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  A <s
B )  ->  { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  =/=  (/) )
45 oninton 6585 . 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 667 1  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  A <s
B )  ->  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  On )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    \/ wo 369    /\ wa 370    = wceq 1437    e. wcel 1872    =/= wne 2599   A.wral 2714   E.wrex 2715   {crab 2718    C_ wss 3379   (/)c0 3704   |^|cint 4198   class class class wbr 4366   Ord word 5384   Oncon0 5385    Fn wfn 5539   ` cfv 5544   Nocsur 30478   <scslt 30479   bdaycbday 30480
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-rep 4479  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-ral 2719  df-rex 2720  df-reu 2721  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-pss 3395  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-tp 3946  df-op 3948  df-uni 4163  df-int 4199  df-iun 4244  df-br 4367  df-opab 4426  df-mpt 4427  df-tr 4462  df-eprel 4707  df-id 4711  df-po 4717  df-so 4718  df-fr 4755  df-we 4757  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-ord 5388  df-on 5389  df-suc 5391  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-f1 5549  df-fo 5550  df-f1o 5551  df-fv 5552  df-1o 7137  df-2o 7138  df-no 30481  df-slt 30482  df-bday 30483
This theorem is referenced by:  nodenselem5  30523  nodenselem6  30524  nodenselem7  30525  nodense  30527
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