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Theorem nodenselem4 27847
Description: Lemma for nodense 27852. 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 3458 . 2  |-  { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  C_  On
2 sltirr 27833 . . . . . . 7  |-  ( A  e.  No  ->  -.  A <s A )
3 breq2 4317 . . . . . . . . 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 27815 . . . . . . . 8  |-  ( A  e.  No  ->  A  Fn  ( bday `  A
) )
9 nofnbday 27815 . . . . . . . 8  |-  ( B  e.  No  ->  B  Fn  ( bday `  B
) )
10 eqfnfv2 5819 . . . . . . . 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 27843 . . . . . . . . . . . 12  |-  ( bday `  A )  e.  On
1514onordi 4844 . . . . . . . . . . 11  |-  Ord  ( bday `  A )
16 bdayelon 27843 . . . . . . . . . . . 12  |-  ( bday `  B )  e.  On
1716onordi 4844 . . . . . . . . . . 11  |-  Ord  ( bday `  B )
18 ordtri3 4776 . . . . . . . . . . 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 27846 . . . . . . . . . 10  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( ( bday `  A
)  e.  ( bday `  B )  ->  E. a  e.  On  ( A `  a )  =/=  ( B `  a )
) )
22 nodenselem3 27846 . . . . . . . . . . . 12  |-  ( ( B  e.  No  /\  A  e.  No )  ->  ( ( bday `  B
)  e.  ( bday `  A )  ->  E. a  e.  On  ( B `  a )  =/=  ( A `  a )
) )
23 necom 2638 . . . . . . . . . . . . 13  |-  ( ( B `  a )  =/=  ( A `  a )  <->  ( A `  a )  =/=  ( B `  a )
)
2423rexbii 2761 . . . . . . . . . . . 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 2747 . . . . . . . . . 10  |-  ( E. a  e.  ( bday `  A )  -.  ( A `  a )  =  ( B `  a )  <->  -.  A. a  e.  ( bday `  A
) ( A `  a )  =  ( B `  a ) )
3014onssi 6469 . . . . . . . . . . . 12  |-  ( bday `  A )  C_  On
31 ssrexv 3438 . . . . . . . . . . . 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 2622 . . . . . . . . . . . 12  |-  ( ( A `  a )  =/=  ( B `  a )  <->  -.  ( A `  a )  =  ( B `  a ) )
3433rexbii 2761 . . . . . . . . . . 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 3678 . . 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 6432 . 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 1369    e. wcel 1756    =/= wne 2620   A.wral 2736   E.wrex 2737   {crab 2740    C_ wss 3349   (/)c0 3658   |^|cint 4149   class class class wbr 4313   Ord word 4739   Oncon0 4740    Fn wfn 5434   ` cfv 5439   Nocsur 27803   <scslt 27804   bdaycbday 27805
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-int 4150  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-on 4744  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-1o 6941  df-2o 6942  df-no 27806  df-slt 27807  df-bday 27808
This theorem is referenced by:  nodenselem5  27848  nodenselem6  27849  nodenselem7  27850  nodense  27852
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