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Theorem alephdom 8474
Description: Relationship between inclusion of ordinal numbers and dominance of infinite initial ordinals. (Contributed by Jeff Hankins, 23-Oct-2009.)
Assertion
Ref Expression
alephdom  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  C_  B  <->  (
aleph `  A )  ~<_  (
aleph `  B ) ) )

Proof of Theorem alephdom
StepHypRef Expression
1 onsseleq 4925 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  C_  B  <->  ( A  e.  B  \/  A  =  B )
) )
2 alephord 8468 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  e.  B  <->  (
aleph `  A )  ~< 
( aleph `  B )
) )
3 sdomdom 7555 . . . . 5  |-  ( (
aleph `  A )  ~< 
( aleph `  B )  ->  ( aleph `  A )  ~<_  ( aleph `  B )
)
42, 3syl6bi 228 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  e.  B  ->  ( aleph `  A )  ~<_  ( aleph `  B )
) )
5 fvex 5882 . . . . . . 7  |-  ( aleph `  A )  e.  _V
6 fveq2 5872 . . . . . . 7  |-  ( A  =  B  ->  ( aleph `  A )  =  ( aleph `  B )
)
7 eqeng 7561 . . . . . . 7  |-  ( (
aleph `  A )  e. 
_V  ->  ( ( aleph `  A )  =  (
aleph `  B )  -> 
( aleph `  A )  ~~  ( aleph `  B )
) )
85, 6, 7mpsyl 63 . . . . . 6  |-  ( A  =  B  ->  ( aleph `  A )  ~~  ( aleph `  B )
)
98a1i 11 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  =  B  ->  ( aleph `  A
)  ~~  ( aleph `  B ) ) )
10 endom 7554 . . . . 5  |-  ( (
aleph `  A )  ~~  ( aleph `  B )  ->  ( aleph `  A )  ~<_  ( aleph `  B )
)
119, 10syl6 33 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  =  B  ->  ( aleph `  A
)  ~<_  ( aleph `  B
) ) )
124, 11jaod 380 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( A  e.  B  \/  A  =  B )  ->  ( aleph `  A )  ~<_  (
aleph `  B ) ) )
131, 12sylbid 215 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  C_  B  ->  ( aleph `  A )  ~<_  ( aleph `  B )
) )
14 eloni 4894 . . . . . 6  |-  ( B  e.  On  ->  Ord  B )
15 eloni 4894 . . . . . 6  |-  ( A  e.  On  ->  Ord  A )
16 ordtri2or 4979 . . . . . 6  |-  ( ( Ord  B  /\  Ord  A )  ->  ( B  e.  A  \/  A  C_  B ) )
1714, 15, 16syl2anr 478 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( B  e.  A  \/  A  C_  B ) )
1817ord 377 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( -.  B  e.  A  ->  A  C_  B
) )
1918con1d 124 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( -.  A  C_  B  ->  B  e.  A
) )
20 alephord 8468 . . . . 5  |-  ( ( B  e.  On  /\  A  e.  On )  ->  ( B  e.  A  <->  (
aleph `  B )  ~< 
( aleph `  A )
) )
2120ancoms 453 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( B  e.  A  <->  (
aleph `  B )  ~< 
( aleph `  A )
) )
22 sdomnen 7556 . . . . 5  |-  ( (
aleph `  B )  ~< 
( aleph `  A )  ->  -.  ( aleph `  B
)  ~~  ( aleph `  A ) )
23 sdomdom 7555 . . . . . 6  |-  ( (
aleph `  B )  ~< 
( aleph `  A )  ->  ( aleph `  B )  ~<_  ( aleph `  A )
)
24 sbth 7649 . . . . . . 7  |-  ( ( ( aleph `  B )  ~<_  ( aleph `  A )  /\  ( aleph `  A )  ~<_  ( aleph `  B )
)  ->  ( aleph `  B )  ~~  ( aleph `  A ) )
2524ex 434 . . . . . 6  |-  ( (
aleph `  B )  ~<_  (
aleph `  A )  -> 
( ( aleph `  A
)  ~<_  ( aleph `  B
)  ->  ( aleph `  B )  ~~  ( aleph `  A ) ) )
2623, 25syl 16 . . . . 5  |-  ( (
aleph `  B )  ~< 
( aleph `  A )  ->  ( ( aleph `  A
)  ~<_  ( aleph `  B
)  ->  ( aleph `  B )  ~~  ( aleph `  A ) ) )
2722, 26mtod 177 . . . 4  |-  ( (
aleph `  B )  ~< 
( aleph `  A )  ->  -.  ( aleph `  A
)  ~<_  ( aleph `  B
) )
2821, 27syl6bi 228 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( B  e.  A  ->  -.  ( aleph `  A
)  ~<_  ( aleph `  B
) ) )
2919, 28syld 44 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( -.  A  C_  B  ->  -.  ( aleph `  A )  ~<_  ( aleph `  B ) ) )
3013, 29impcon4bid 205 1  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  C_  B  <->  (
aleph `  A )  ~<_  (
aleph `  B ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3118    C_ wss 3481   class class class wbr 4453   Ord word 4883   Oncon0 4884   ` cfv 5594    ~~ cen 7525    ~<_ cdom 7526    ~< csdm 7527   alephcale 8329
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  ax-inf2 8070
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-rmo 2825  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-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  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-isom 5603  df-riota 6256  df-om 6696  df-recs 7054  df-rdg 7088  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-oi 7947  df-har 7996  df-card 8332  df-aleph 8333
This theorem is referenced by: (None)
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