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Theorem alephord 8504
Description: Ordering property of the aleph function. (Contributed by NM, 26-Oct-2003.) (Revised by Mario Carneiro, 9-Feb-2013.)
Assertion
Ref Expression
alephord  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  e.  B  <->  (
aleph `  A )  ~< 
( aleph `  B )
) )

Proof of Theorem alephord
StepHypRef Expression
1 alephordi 8503 . . 3  |-  ( B  e.  On  ->  ( A  e.  B  ->  (
aleph `  A )  ~< 
( aleph `  B )
) )
21adantl 467 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  e.  B  ->  ( aleph `  A )  ~<  ( aleph `  B )
) )
3 brsdom 7599 . . 3  |-  ( (
aleph `  A )  ~< 
( aleph `  B )  <->  ( ( aleph `  A )  ~<_  ( aleph `  B )  /\  -.  ( aleph `  A
)  ~~  ( aleph `  B ) ) )
4 alephon 8498 . . . . . . . . 9  |-  ( aleph `  A )  e.  On
5 alephon 8498 . . . . . . . . 9  |-  ( aleph `  B )  e.  On
6 domtriord 7724 . . . . . . . . 9  |-  ( ( ( aleph `  A )  e.  On  /\  ( aleph `  B )  e.  On )  ->  ( ( aleph `  A )  ~<_  ( aleph `  B )  <->  -.  ( aleph `  B )  ~< 
( aleph `  A )
) )
74, 5, 6mp2an 676 . . . . . . . 8  |-  ( (
aleph `  A )  ~<_  (
aleph `  B )  <->  -.  ( aleph `  B )  ~< 
( aleph `  A )
)
8 alephordi 8503 . . . . . . . . 9  |-  ( A  e.  On  ->  ( B  e.  A  ->  (
aleph `  B )  ~< 
( aleph `  A )
) )
98con3d 138 . . . . . . . 8  |-  ( A  e.  On  ->  ( -.  ( aleph `  B )  ~<  ( aleph `  A )  ->  -.  B  e.  A
) )
107, 9syl5bi 220 . . . . . . 7  |-  ( A  e.  On  ->  (
( aleph `  A )  ~<_  ( aleph `  B )  ->  -.  B  e.  A
) )
1110adantr 466 . . . . . 6  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( aleph `  A
)  ~<_  ( aleph `  B
)  ->  -.  B  e.  A ) )
12 ontri1 5476 . . . . . 6  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  C_  B  <->  -.  B  e.  A ) )
1311, 12sylibrd 237 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( aleph `  A
)  ~<_  ( aleph `  B
)  ->  A  C_  B
) )
14 fveq2 5881 . . . . . . . 8  |-  ( A  =  B  ->  ( aleph `  A )  =  ( aleph `  B )
)
15 eqeng 7610 . . . . . . . 8  |-  ( (
aleph `  A )  e.  On  ->  ( ( aleph `  A )  =  ( aleph `  B )  ->  ( aleph `  A )  ~~  ( aleph `  B )
) )
164, 14, 15mpsyl 65 . . . . . . 7  |-  ( A  =  B  ->  ( aleph `  A )  ~~  ( aleph `  B )
)
1716necon3bi 2660 . . . . . 6  |-  ( -.  ( aleph `  A )  ~~  ( aleph `  B )  ->  A  =/=  B )
1817a1i 11 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( -.  ( aleph `  A )  ~~  ( aleph `  B )  ->  A  =/=  B ) )
1913, 18anim12d 565 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( ( aleph `  A )  ~<_  ( aleph `  B )  /\  -.  ( aleph `  A )  ~~  ( aleph `  B )
)  ->  ( A  C_  B  /\  A  =/= 
B ) ) )
20 onelpss 5482 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  e.  B  <->  ( A  C_  B  /\  A  =/=  B ) ) )
2119, 20sylibrd 237 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( ( aleph `  A )  ~<_  ( aleph `  B )  /\  -.  ( aleph `  A )  ~~  ( aleph `  B )
)  ->  A  e.  B ) )
223, 21syl5bi 220 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( aleph `  A
)  ~<  ( aleph `  B
)  ->  A  e.  B ) )
232, 22impbid 193 1  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  e.  B  <->  (
aleph `  A )  ~< 
( aleph `  B )
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1870    =/= wne 2625    C_ wss 3442   class class class wbr 4426   Oncon0 5442   ` cfv 5601    ~~ cen 7574    ~<_ cdom 7575    ~< csdm 7576   alephcale 8369
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-inf2 8146
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 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-se 4814  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6267  df-om 6707  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-oi 8025  df-har 8073  df-card 8372  df-aleph 8373
This theorem is referenced by:  alephord2  8505  alephdom  8510  alephval2  8995
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