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Theorem alephord 7956
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 7955 . . 3  |-  ( B  e.  On  ->  ( A  e.  B  ->  (
aleph `  A )  ~< 
( aleph `  B )
) )
21adantl 453 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  e.  B  ->  ( aleph `  A )  ~<  ( aleph `  B )
) )
3 brsdom 7130 . . 3  |-  ( (
aleph `  A )  ~< 
( aleph `  B )  <->  ( ( aleph `  A )  ~<_  ( aleph `  B )  /\  -.  ( aleph `  A
)  ~~  ( aleph `  B ) ) )
4 alephon 7950 . . . . . . . . 9  |-  ( aleph `  A )  e.  On
5 alephon 7950 . . . . . . . . 9  |-  ( aleph `  B )  e.  On
6 domtriord 7253 . . . . . . . . 9  |-  ( ( ( aleph `  A )  e.  On  /\  ( aleph `  B )  e.  On )  ->  ( ( aleph `  A )  ~<_  ( aleph `  B )  <->  -.  ( aleph `  B )  ~< 
( aleph `  A )
) )
74, 5, 6mp2an 654 . . . . . . . 8  |-  ( (
aleph `  A )  ~<_  (
aleph `  B )  <->  -.  ( aleph `  B )  ~< 
( aleph `  A )
)
8 alephordi 7955 . . . . . . . . 9  |-  ( A  e.  On  ->  ( B  e.  A  ->  (
aleph `  B )  ~< 
( aleph `  A )
) )
98con3d 127 . . . . . . . 8  |-  ( A  e.  On  ->  ( -.  ( aleph `  B )  ~<  ( aleph `  A )  ->  -.  B  e.  A
) )
107, 9syl5bi 209 . . . . . . 7  |-  ( A  e.  On  ->  (
( aleph `  A )  ~<_  ( aleph `  B )  ->  -.  B  e.  A
) )
1110adantr 452 . . . . . 6  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( aleph `  A
)  ~<_  ( aleph `  B
)  ->  -.  B  e.  A ) )
12 ontri1 4615 . . . . . 6  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  C_  B  <->  -.  B  e.  A ) )
1311, 12sylibrd 226 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( aleph `  A
)  ~<_  ( aleph `  B
)  ->  A  C_  B
) )
14 fveq2 5728 . . . . . . . 8  |-  ( A  =  B  ->  ( aleph `  A )  =  ( aleph `  B )
)
15 eqeng 7141 . . . . . . . 8  |-  ( (
aleph `  A )  e.  On  ->  ( ( aleph `  A )  =  ( aleph `  B )  ->  ( aleph `  A )  ~~  ( aleph `  B )
) )
164, 14, 15mpsyl 61 . . . . . . 7  |-  ( A  =  B  ->  ( aleph `  A )  ~~  ( aleph `  B )
)
1716necon3bi 2645 . . . . . 6  |-  ( -.  ( aleph `  A )  ~~  ( aleph `  B )  ->  A  =/=  B )
1817a1i 11 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( -.  ( aleph `  A )  ~~  ( aleph `  B )  ->  A  =/=  B ) )
1913, 18anim12d 547 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( ( aleph `  A )  ~<_  ( aleph `  B )  /\  -.  ( aleph `  A )  ~~  ( aleph `  B )
)  ->  ( A  C_  B  /\  A  =/= 
B ) ) )
20 onelpss 4621 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  e.  B  <->  ( A  C_  B  /\  A  =/=  B ) ) )
2119, 20sylibrd 226 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( ( aleph `  A )  ~<_  ( aleph `  B )  /\  -.  ( aleph `  A )  ~~  ( aleph `  B )
)  ->  A  e.  B ) )
223, 21syl5bi 209 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( aleph `  A
)  ~<  ( aleph `  B
)  ->  A  e.  B ) )
232, 22impbid 184 1  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  e.  B  <->  (
aleph `  A )  ~< 
( aleph `  B )
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2599    C_ wss 3320   class class class wbr 4212   Oncon0 4581   ` cfv 5454    ~~ cen 7106    ~<_ cdom 7107    ~< csdm 7108   alephcale 7823
This theorem is referenced by:  alephord2  7957  alephdom  7962  alephval2  8447
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-oi 7479  df-har 7526  df-card 7826  df-aleph 7827
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