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Theorem alephdom 8512
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 5479 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  C_  B  <->  ( A  e.  B  \/  A  =  B )
) )
2 alephord 8506 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  e.  B  <->  (
aleph `  A )  ~< 
( aleph `  B )
) )
3 sdomdom 7600 . . . . 5  |-  ( (
aleph `  A )  ~< 
( aleph `  B )  ->  ( aleph `  A )  ~<_  ( aleph `  B )
)
42, 3syl6bi 231 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  e.  B  ->  ( aleph `  A )  ~<_  ( aleph `  B )
) )
5 fvex 5887 . . . . . . 7  |-  ( aleph `  A )  e.  _V
6 fveq2 5877 . . . . . . 7  |-  ( A  =  B  ->  ( aleph `  A )  =  ( aleph `  B )
)
7 eqeng 7606 . . . . . . 7  |-  ( (
aleph `  A )  e. 
_V  ->  ( ( aleph `  A )  =  (
aleph `  B )  -> 
( aleph `  A )  ~~  ( aleph `  B )
) )
85, 6, 7mpsyl 65 . . . . . 6  |-  ( A  =  B  ->  ( aleph `  A )  ~~  ( aleph `  B )
)
98a1i 11 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  =  B  ->  ( aleph `  A
)  ~~  ( aleph `  B ) ) )
10 endom 7599 . . . . 5  |-  ( (
aleph `  A )  ~~  ( aleph `  B )  ->  ( aleph `  A )  ~<_  ( aleph `  B )
)
119, 10syl6 34 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  =  B  ->  ( aleph `  A
)  ~<_  ( aleph `  B
) ) )
124, 11jaod 381 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( A  e.  B  \/  A  =  B )  ->  ( aleph `  A )  ~<_  (
aleph `  B ) ) )
131, 12sylbid 218 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  C_  B  ->  ( aleph `  A )  ~<_  ( aleph `  B )
) )
14 eloni 5448 . . . . . 6  |-  ( B  e.  On  ->  Ord  B )
15 eloni 5448 . . . . . 6  |-  ( A  e.  On  ->  Ord  A )
16 ordtri2or 5533 . . . . . 6  |-  ( ( Ord  B  /\  Ord  A )  ->  ( B  e.  A  \/  A  C_  B ) )
1714, 15, 16syl2anr 480 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( B  e.  A  \/  A  C_  B ) )
1817ord 378 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( -.  B  e.  A  ->  A  C_  B
) )
1918con1d 127 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( -.  A  C_  B  ->  B  e.  A
) )
20 alephord 8506 . . . . 5  |-  ( ( B  e.  On  /\  A  e.  On )  ->  ( B  e.  A  <->  (
aleph `  B )  ~< 
( aleph `  A )
) )
2120ancoms 454 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( B  e.  A  <->  (
aleph `  B )  ~< 
( aleph `  A )
) )
22 sdomnen 7601 . . . . 5  |-  ( (
aleph `  B )  ~< 
( aleph `  A )  ->  -.  ( aleph `  B
)  ~~  ( aleph `  A ) )
23 sdomdom 7600 . . . . . 6  |-  ( (
aleph `  B )  ~< 
( aleph `  A )  ->  ( aleph `  B )  ~<_  ( aleph `  A )
)
24 sbth 7694 . . . . . . 7  |-  ( ( ( aleph `  B )  ~<_  ( aleph `  A )  /\  ( aleph `  A )  ~<_  ( aleph `  B )
)  ->  ( aleph `  B )  ~~  ( aleph `  A ) )
2524ex 435 . . . . . 6  |-  ( (
aleph `  B )  ~<_  (
aleph `  A )  -> 
( ( aleph `  A
)  ~<_  ( aleph `  B
)  ->  ( aleph `  B )  ~~  ( aleph `  A ) ) )
2623, 25syl 17 . . . . 5  |-  ( (
aleph `  B )  ~< 
( aleph `  A )  ->  ( ( aleph `  A
)  ~<_  ( aleph `  B
)  ->  ( aleph `  B )  ~~  ( aleph `  A ) ) )
2722, 26mtod 180 . . . 4  |-  ( (
aleph `  B )  ~< 
( aleph `  A )  ->  -.  ( aleph `  A
)  ~<_  ( aleph `  B
) )
2821, 27syl6bi 231 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( B  e.  A  ->  -.  ( aleph `  A
)  ~<_  ( aleph `  B
) ) )
2919, 28syld 45 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( -.  A  C_  B  ->  -.  ( aleph `  A )  ~<_  ( aleph `  B ) ) )
3013, 29impcon4bid 208 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 187    \/ wo 369    /\ wa 370    = wceq 1437    e. wcel 1868   _Vcvv 3081    C_ wss 3436   class class class wbr 4420   Ord word 5437   Oncon0 5438   ` cfv 5597    ~~ cen 7570    ~<_ cdom 7571    ~< csdm 7572   alephcale 8371
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 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4551  ax-pow 4598  ax-pr 4656  ax-un 6593  ax-inf2 8148
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 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-int 4253  df-iun 4298  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4760  df-id 4764  df-po 4770  df-so 4771  df-fr 4808  df-se 4809  df-we 4810  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-rn 4860  df-res 4861  df-ima 4862  df-pred 5395  df-ord 5441  df-on 5442  df-lim 5443  df-suc 5444  df-iota 5561  df-fun 5599  df-fn 5600  df-f 5601  df-f1 5602  df-fo 5603  df-f1o 5604  df-fv 5605  df-isom 5606  df-riota 6263  df-om 6703  df-wrecs 7032  df-recs 7094  df-rdg 7132  df-er 7367  df-en 7574  df-dom 7575  df-sdom 7576  df-oi 8027  df-har 8075  df-card 8374  df-aleph 8375
This theorem is referenced by: (None)
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