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Theorem nodenselem4 29378
Description: Lemma for nodense 29383. 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 3590 . 2  |-  { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  C_  On
2 sltirr 29364 . . . . . . 7  |-  ( A  e.  No  ->  -.  A <s A )
3 breq2 4457 . . . . . . . . 9  |-  ( A  =  B  ->  ( A <s A  <->  A <s B ) )
43biimprcd 225 . . . . . . . 8  |-  ( A <s B  -> 
( A  =  B  ->  A <s
A ) )
54con3d 133 . . . . . . 7  |-  ( A <s B  -> 
( -.  A <s A  ->  -.  A  =  B ) )
62, 5syl5com 30 . . . . . 6  |-  ( A  e.  No  ->  ( A <s B  ->  -.  A  =  B
) )
76adantr 465 . . . . 5  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( A <s
B  ->  -.  A  =  B ) )
8 nofnbday 29346 . . . . . . . 8  |-  ( A  e.  No  ->  A  Fn  ( bday `  A
) )
9 nofnbday 29346 . . . . . . . 8  |-  ( B  e.  No  ->  B  Fn  ( bday `  B
) )
10 eqfnfv2 5983 . . . . . . . 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 477 . . . . . . 7  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( A  =  B  <-> 
( ( bday `  A
)  =  ( bday `  B )  /\  A. a  e.  ( bday `  A ) ( A `
 a )  =  ( B `  a
) ) ) )
1211notbid 294 . . . . . 6  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( -.  A  =  B  <->  -.  ( ( bday `  A )  =  ( bday `  B
)  /\  A. a  e.  ( bday `  A
) ( A `  a )  =  ( B `  a ) ) ) )
13 ianor 488 . . . . . . 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 29374 . . . . . . . . . . . 12  |-  ( bday `  A )  e.  On
1514onordi 4988 . . . . . . . . . . 11  |-  Ord  ( bday `  A )
16 bdayelon 29374 . . . . . . . . . . . 12  |-  ( bday `  B )  e.  On
1716onordi 4988 . . . . . . . . . . 11  |-  Ord  ( bday `  B )
18 ordtri3 4920 . . . . . . . . . . 11  |-  ( ( Ord  ( bday `  A
)  /\  Ord  ( bday `  B ) )  -> 
( ( bday `  A
)  =  ( bday `  B )  <->  -.  (
( bday `  A )  e.  ( bday `  B
)  \/  ( bday `  B )  e.  (
bday `  A )
) ) )
1915, 17, 18mp2an 672 . . . . . . . . . 10  |-  ( (
bday `  A )  =  ( bday `  B
)  <->  -.  ( ( bday `  A )  e.  ( bday `  B
)  \/  ( bday `  B )  e.  (
bday `  A )
) )
2019con2bii 332 . . . . . . . . 9  |-  ( ( ( bday `  A
)  e.  ( bday `  B )  \/  ( bday `  B )  e.  ( bday `  A
) )  <->  -.  ( bday `  A )  =  ( bday `  B
) )
21 nodenselem3 29377 . . . . . . . . . 10  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( ( bday `  A
)  e.  ( bday `  B )  ->  E. a  e.  On  ( A `  a )  =/=  ( B `  a )
) )
22 nodenselem3 29377 . . . . . . . . . . . 12  |-  ( ( B  e.  No  /\  A  e.  No )  ->  ( ( bday `  B
)  e.  ( bday `  A )  ->  E. a  e.  On  ( B `  a )  =/=  ( A `  a )
) )
23 necom 2736 . . . . . . . . . . . . 13  |-  ( ( B `  a )  =/=  ( A `  a )  <->  ( A `  a )  =/=  ( B `  a )
)
2423rexbii 2969 . . . . . . . . . . . 12  |-  ( E. a  e.  On  ( B `  a )  =/=  ( A `  a
)  <->  E. a  e.  On  ( A `  a )  =/=  ( B `  a ) )
2522, 24syl6ib 226 . . . . . . . . . . 11  |-  ( ( B  e.  No  /\  A  e.  No )  ->  ( ( bday `  B
)  e.  ( bday `  A )  ->  E. a  e.  On  ( A `  a )  =/=  ( B `  a )
) )
2625ancoms 453 . . . . . . . . . 10  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( ( bday `  B
)  e.  ( bday `  A )  ->  E. a  e.  On  ( A `  a )  =/=  ( B `  a )
) )
2721, 26jaod 380 . . . . . . . . 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 218 . . . . . . . 8  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( -.  ( bday `  A )  =  (
bday `  B )  ->  E. a  e.  On  ( A `  a )  =/=  ( B `  a ) ) )
29 rexnal 2915 . . . . . . . . . 10  |-  ( E. a  e.  ( bday `  A )  -.  ( A `  a )  =  ( B `  a )  <->  -.  A. a  e.  ( bday `  A
) ( A `  a )  =  ( B `  a ) )
3014onssi 6667 . . . . . . . . . . . 12  |-  ( bday `  A )  C_  On
31 ssrexv 3570 . . . . . . . . . . . 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 2664 . . . . . . . . . . . 12  |-  ( ( A `  a )  =/=  ( B `  a )  <->  -.  ( A `  a )  =  ( B `  a ) )
3433rexbii 2969 . . . . . . . . . . 11  |-  ( E. a  e.  On  ( A `  a )  =/=  ( B `  a
)  <->  E. a  e.  On  -.  ( A `  a
)  =  ( B `
 a ) )
3532, 34sylibr 212 . . . . . . . . . 10  |-  ( E. a  e.  ( bday `  A )  -.  ( A `  a )  =  ( B `  a )  ->  E. a  e.  On  ( A `  a )  =/=  ( B `  a )
)
3629, 35sylbir 213 . . . . . . . . 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 380 . . . . . . 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 217 . . . . . 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 215 . . . . 5  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( -.  A  =  B  ->  E. a  e.  On  ( A `  a )  =/=  ( B `  a )
) )
417, 40syld 44 . . . 4  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( A <s
B  ->  E. a  e.  On  ( A `  a )  =/=  ( B `  a )
) )
4241imp 429 . . 3  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  A <s
B )  ->  E. a  e.  On  ( A `  a )  =/=  ( B `  a )
)
43 rabn0 3810 . . 3  |-  ( { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) }  =/=  (/)  <->  E. a  e.  On  ( A `  a )  =/=  ( B `  a ) )
4442, 43sylibr 212 . 2  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  A <s
B )  ->  { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  =/=  (/) )
45 oninton 6630 . 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 663 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 184    \/ wo 368    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2817   E.wrex 2818   {crab 2821    C_ wss 3481   (/)c0 3790   |^|cint 4288   class class class wbr 4453   Ord word 4883   Oncon0 4884    Fn wfn 5589   ` cfv 5594   Nocsur 29334   <scslt 29335   bdaycbday 29336
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  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 7142  df-2o 7143  df-no 29337  df-slt 29338  df-bday 29339
This theorem is referenced by:  nodenselem5  29379  nodenselem6  29380  nodenselem7  29381  nodense  29383
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