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Theorem nodenselem4 24723
Description: Lemma for nodense 24728. 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 3292 . 2  |-  { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  C_  On
2 sltirr 24709 . . . . . . 7  |-  ( A  e.  No  ->  -.  A < s A )
3 breq2 4064 . . . . . . . . 9  |-  ( A  =  B  ->  ( A < s A  <->  A < s B ) )
43biimprcd 216 . . . . . . . 8  |-  ( A < s B  -> 
( A  =  B  ->  A < s A ) )
54con3d 125 . . . . . . 7  |-  ( A < s B  -> 
( -.  A <
s A  ->  -.  A  =  B )
)
62, 5syl5com 26 . . . . . 6  |-  ( A  e.  No  ->  ( A < s B  ->  -.  A  =  B
) )
76adantr 451 . . . . 5  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( A < s B  ->  -.  A  =  B ) )
8 nofnbday 24691 . . . . . . . 8  |-  ( A  e.  No  ->  A  Fn  ( bday `  A
) )
9 nofnbday 24691 . . . . . . . 8  |-  ( B  e.  No  ->  B  Fn  ( bday `  B
) )
10 eqfnfv2 5661 . . . . . . . 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 463 . . . . . . 7  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( A  =  B  <-> 
( ( bday `  A
)  =  ( bday `  B )  /\  A. a  e.  ( bday `  A ) ( A `
 a )  =  ( B `  a
) ) ) )
1211notbid 285 . . . . . 6  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( -.  A  =  B  <->  -.  ( ( bday `  A )  =  ( bday `  B
)  /\  A. a  e.  ( bday `  A
) ( A `  a )  =  ( B `  a ) ) ) )
13 ianor 474 . . . . . . 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 24719 . . . . . . . . . . . 12  |-  ( bday `  A )  e.  On
1514onordi 4534 . . . . . . . . . . 11  |-  Ord  ( bday `  A )
16 bdayelon 24719 . . . . . . . . . . . 12  |-  ( bday `  B )  e.  On
1716onordi 4534 . . . . . . . . . . 11  |-  Ord  ( bday `  B )
18 ordtri3 4465 . . . . . . . . . . 11  |-  ( ( Ord  ( bday `  A
)  /\  Ord  ( bday `  B ) )  -> 
( ( bday `  A
)  =  ( bday `  B )  <->  -.  (
( bday `  A )  e.  ( bday `  B
)  \/  ( bday `  B )  e.  (
bday `  A )
) ) )
1915, 17, 18mp2an 653 . . . . . . . . . 10  |-  ( (
bday `  A )  =  ( bday `  B
)  <->  -.  ( ( bday `  A )  e.  ( bday `  B
)  \/  ( bday `  B )  e.  (
bday `  A )
) )
2019con2bii 322 . . . . . . . . 9  |-  ( ( ( bday `  A
)  e.  ( bday `  B )  \/  ( bday `  B )  e.  ( bday `  A
) )  <->  -.  ( bday `  A )  =  ( bday `  B
) )
21 nodenselem3 24722 . . . . . . . . . 10  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( ( bday `  A
)  e.  ( bday `  B )  ->  E. a  e.  On  ( A `  a )  =/=  ( B `  a )
) )
22 nodenselem3 24722 . . . . . . . . . . . 12  |-  ( ( B  e.  No  /\  A  e.  No )  ->  ( ( bday `  B
)  e.  ( bday `  A )  ->  E. a  e.  On  ( B `  a )  =/=  ( A `  a )
) )
23 necom 2560 . . . . . . . . . . . . 13  |-  ( ( B `  a )  =/=  ( A `  a )  <->  ( A `  a )  =/=  ( B `  a )
)
2423rexbii 2602 . . . . . . . . . . . 12  |-  ( E. a  e.  On  ( B `  a )  =/=  ( A `  a
)  <->  E. a  e.  On  ( A `  a )  =/=  ( B `  a ) )
2522, 24syl6ib 217 . . . . . . . . . . 11  |-  ( ( B  e.  No  /\  A  e.  No )  ->  ( ( bday `  B
)  e.  ( bday `  A )  ->  E. a  e.  On  ( A `  a )  =/=  ( B `  a )
) )
2625ancoms 439 . . . . . . . . . 10  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( ( bday `  B
)  e.  ( bday `  A )  ->  E. a  e.  On  ( A `  a )  =/=  ( B `  a )
) )
2721, 26jaod 369 . . . . . . . . 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 209 . . . . . . . 8  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( -.  ( bday `  A )  =  (
bday `  B )  ->  E. a  e.  On  ( A `  a )  =/=  ( B `  a ) ) )
29 rexnal 2588 . . . . . . . . . 10  |-  ( E. a  e.  ( bday `  A )  -.  ( A `  a )  =  ( B `  a )  <->  -.  A. a  e.  ( bday `  A
) ( A `  a )  =  ( B `  a ) )
3014onssi 4665 . . . . . . . . . . . 12  |-  ( bday `  A )  C_  On
31 ssrexv 3272 . . . . . . . . . . . 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 2481 . . . . . . . . . . . 12  |-  ( ( A `  a )  =/=  ( B `  a )  <->  -.  ( A `  a )  =  ( B `  a ) )
3433rexbii 2602 . . . . . . . . . . 11  |-  ( E. a  e.  On  ( A `  a )  =/=  ( B `  a
)  <->  E. a  e.  On  -.  ( A `  a
)  =  ( B `
 a ) )
3532, 34sylibr 203 . . . . . . . . . 10  |-  ( E. a  e.  ( bday `  A )  -.  ( A `  a )  =  ( B `  a )  ->  E. a  e.  On  ( A `  a )  =/=  ( B `  a )
)
3629, 35sylbir 204 . . . . . . . . 9  |-  ( -. 
A. a  e.  (
bday `  A )
( A `  a
)  =  ( B `
 a )  ->  E. a  e.  On  ( A `  a )  =/=  ( B `  a ) )
3736a1i 10 . . . . . . . 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 369 . . . . . . 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 208 . . . . . 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 206 . . . . 5  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( -.  A  =  B  ->  E. a  e.  On  ( A `  a )  =/=  ( B `  a )
) )
417, 40syld 40 . . . 4  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( A < s B  ->  E. a  e.  On  ( A `  a )  =/=  ( B `  a ) ) )
4241imp 418 . . 3  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  A < s B )  ->  E. a  e.  On  ( A `  a )  =/=  ( B `  a )
)
43 rabn0 3508 . . 3  |-  ( { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) }  =/=  (/)  <->  E. a  e.  On  ( A `  a )  =/=  ( B `  a ) )
4442, 43sylibr 203 . 2  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  A < s B )  ->  { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  =/=  (/) )
45 oninton 4628 . 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 644 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 176    \/ wo 357    /\ wa 358    = wceq 1633    e. wcel 1701    =/= wne 2479   A.wral 2577   E.wrex 2578   {crab 2581    C_ wss 3186   (/)c0 3489   |^|cint 3899   class class class wbr 4060   Ord word 4428   Oncon0 4429    Fn wfn 5287   ` cfv 5292   Nocsur 24679   < scslt 24680   bdaycbday 24681
This theorem is referenced by:  nodenselem5  24724  nodenselem6  24725  nodenselem7  24726  nodense  24728
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-int 3900  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-suc 4435  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-1o 6521  df-2o 6522  df-no 24682  df-slt 24683  df-bday 24684
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