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Theorem alephdom 7798
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 4515 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  C_  B  <->  ( A  e.  B  \/  A  =  B )
) )
2 alephord 7792 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  e.  B  <->  (
aleph `  A )  ~< 
( aleph `  B )
) )
3 sdomdom 6977 . . . . 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 5622 . . . . . . 7  |-  ( aleph `  A )  e.  _V
6 fveq2 5608 . . . . . . 7  |-  ( A  =  B  ->  ( aleph `  A )  =  ( aleph `  B )
)
7 eqeng 6983 . . . . . . 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 6976 . . . . 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 4484 . . . . . 6  |-  ( B  e.  On  ->  Ord  B )
15 eloni 4484 . . . . . 6  |-  ( A  e.  On  ->  Ord  A )
16 ordtri2or 4570 . . . . . 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 7792 . . . . 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 6978 . . . . 5  |-  ( (
aleph `  B )  ~< 
( aleph `  A )  ->  -.  ( aleph `  B
)  ~~  ( aleph `  A ) )
23 sdomdom 6977 . . . . . 6  |-  ( (
aleph `  B )  ~< 
( aleph `  A )  ->  ( aleph `  B )  ~<_  ( aleph `  A )
)
24 sbth 7069 . . . . . . 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 1642    e. wcel 1710   _Vcvv 2864    C_ wss 3228   class class class wbr 4104   Ord word 4473   Oncon0 4474   ` cfv 5337    ~~ cen 6948    ~<_ cdom 6949    ~< csdm 6950   alephcale 7659
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-inf2 7432
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-int 3944  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-se 4435  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-isom 5346  df-riota 6391  df-recs 6475  df-rdg 6510  df-er 6747  df-en 6952  df-dom 6953  df-sdom 6954  df-oi 7315  df-har 7362  df-card 7662  df-aleph 7663
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