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Theorem alephdom 7708
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 4433 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  C_  B  <->  ( A  e.  B  \/  A  =  B )
) )
2 alephord 7702 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  e.  B  <->  (
aleph `  A )  ~< 
( aleph `  B )
) )
3 sdomdom 6889 . . . . 5  |-  ( (
aleph `  A )  ~< 
( aleph `  B )  ->  ( aleph `  A )  ~<_  ( aleph `  B )
)
42, 3syl6bi 219 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  e.  B  ->  ( aleph `  A )  ~<_  ( aleph `  B )
) )
5 fvex 5539 . . . . . . 7  |-  ( aleph `  A )  e.  _V
6 fveq2 5525 . . . . . . 7  |-  ( A  =  B  ->  ( aleph `  A )  =  ( aleph `  B )
)
7 eqeng 6895 . . . . . . 7  |-  ( (
aleph `  A )  e. 
_V  ->  ( ( aleph `  A )  =  (
aleph `  B )  -> 
( aleph `  A )  ~~  ( aleph `  B )
) )
85, 6, 7mpsyl 59 . . . . . 6  |-  ( A  =  B  ->  ( aleph `  A )  ~~  ( aleph `  B )
)
98a1i 10 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  =  B  ->  ( aleph `  A
)  ~~  ( aleph `  B ) ) )
10 endom 6888 . . . . 5  |-  ( (
aleph `  A )  ~~  ( aleph `  B )  ->  ( aleph `  A )  ~<_  ( aleph `  B )
)
119, 10syl6 29 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  =  B  ->  ( aleph `  A
)  ~<_  ( aleph `  B
) ) )
124, 11jaod 369 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( A  e.  B  \/  A  =  B )  ->  ( aleph `  A )  ~<_  (
aleph `  B ) ) )
131, 12sylbid 206 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  C_  B  ->  ( aleph `  A )  ~<_  ( aleph `  B )
) )
14 eloni 4402 . . . . . 6  |-  ( B  e.  On  ->  Ord  B )
15 eloni 4402 . . . . . 6  |-  ( A  e.  On  ->  Ord  A )
16 ordtri2or 4488 . . . . . 6  |-  ( ( Ord  B  /\  Ord  A )  ->  ( B  e.  A  \/  A  C_  B ) )
1714, 15, 16syl2anr 464 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( B  e.  A  \/  A  C_  B ) )
1817ord 366 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( -.  B  e.  A  ->  A  C_  B
) )
1918con1d 116 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( -.  A  C_  B  ->  B  e.  A
) )
20 alephord 7702 . . . . 5  |-  ( ( B  e.  On  /\  A  e.  On )  ->  ( B  e.  A  <->  (
aleph `  B )  ~< 
( aleph `  A )
) )
2120ancoms 439 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( B  e.  A  <->  (
aleph `  B )  ~< 
( aleph `  A )
) )
22 sdomnen 6890 . . . . 5  |-  ( (
aleph `  B )  ~< 
( aleph `  A )  ->  -.  ( aleph `  B
)  ~~  ( aleph `  A ) )
23 sdomdom 6889 . . . . . 6  |-  ( (
aleph `  B )  ~< 
( aleph `  A )  ->  ( aleph `  B )  ~<_  ( aleph `  A )
)
24 sbth 6981 . . . . . . 7  |-  ( ( ( aleph `  B )  ~<_  ( aleph `  A )  /\  ( aleph `  A )  ~<_  ( aleph `  B )
)  ->  ( aleph `  B )  ~~  ( aleph `  A ) )
2524ex 423 . . . . . 6  |-  ( (
aleph `  B )  ~<_  (
aleph `  A )  -> 
( ( aleph `  A
)  ~<_  ( aleph `  B
)  ->  ( aleph `  B )  ~~  ( aleph `  A ) ) )
2623, 25syl 15 . . . . 5  |-  ( (
aleph `  B )  ~< 
( aleph `  A )  ->  ( ( aleph `  A
)  ~<_  ( aleph `  B
)  ->  ( aleph `  B )  ~~  ( aleph `  A ) ) )
2722, 26mtod 168 . . . 4  |-  ( (
aleph `  B )  ~< 
( aleph `  A )  ->  -.  ( aleph `  A
)  ~<_  ( aleph `  B
) )
2821, 27syl6bi 219 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( B  e.  A  ->  -.  ( aleph `  A
)  ~<_  ( aleph `  B
) ) )
2919, 28syld 40 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( -.  A  C_  B  ->  -.  ( aleph `  A )  ~<_  ( aleph `  B ) ) )
3013, 29impcon4bid 196 1  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  C_  B  <->  (
aleph `  A )  ~<_  (
aleph `  B ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788    C_ wss 3152   class class class wbr 4023   Ord word 4391   Oncon0 4392   ` cfv 5255    ~~ cen 6860    ~<_ cdom 6861    ~< csdm 6862   alephcale 7569
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-oi 7225  df-har 7272  df-card 7572  df-aleph 7573
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