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Theorem alephdom2 7928
Description: A dominated initial ordinal is included. (Contributed by Jeff Hankins, 24-Oct-2009.)
Assertion
Ref Expression
alephdom2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( aleph `  A
)  C_  B  <->  ( aleph `  A )  ~<_  B ) )

Proof of Theorem alephdom2
StepHypRef Expression
1 alephsdom 7927 . . . 4  |-  ( ( B  e.  On  /\  A  e.  On )  ->  ( B  e.  (
aleph `  A )  <->  B  ~<  (
aleph `  A ) ) )
21ancoms 440 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( B  e.  (
aleph `  A )  <->  B  ~<  (
aleph `  A ) ) )
32notbid 286 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( -.  B  e.  ( aleph `  A )  <->  -.  B  ~<  ( aleph `  A ) ) )
4 alephon 7910 . . . . 5  |-  ( aleph `  A )  e.  On
54onordi 4649 . . . 4  |-  Ord  ( aleph `  A )
6 eloni 4555 . . . 4  |-  ( B  e.  On  ->  Ord  B )
7 ordtri1 4578 . . . 4  |-  ( ( Ord  ( aleph `  A
)  /\  Ord  B )  ->  ( ( aleph `  A )  C_  B  <->  -.  B  e.  ( aleph `  A ) ) )
85, 6, 7sylancr 645 . . 3  |-  ( B  e.  On  ->  (
( aleph `  A )  C_  B  <->  -.  B  e.  ( aleph `  A )
) )
98adantl 453 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( aleph `  A
)  C_  B  <->  -.  B  e.  ( aleph `  A )
) )
10 domtriord 7216 . . . 4  |-  ( ( ( aleph `  A )  e.  On  /\  B  e.  On )  ->  (
( aleph `  A )  ~<_  B 
<->  -.  B  ~<  ( aleph `  A ) ) )
114, 10mpan 652 . . 3  |-  ( B  e.  On  ->  (
( aleph `  A )  ~<_  B 
<->  -.  B  ~<  ( aleph `  A ) ) )
1211adantl 453 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( aleph `  A
)  ~<_  B  <->  -.  B  ~<  ( aleph `  A )
) )
133, 9, 123bitr4d 277 1  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( aleph `  A
)  C_  B  <->  ( aleph `  A )  ~<_  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    e. wcel 1721    C_ wss 3284   class class class wbr 4176   Ord word 4544   Oncon0 4545   ` cfv 5417    ~<_ cdom 7070    ~< csdm 7071   alephcale 7783
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-inf2 7556
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-int 4015  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-se 4506  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-isom 5426  df-riota 6512  df-recs 6596  df-rdg 6631  df-er 6868  df-en 7073  df-dom 7074  df-sdom 7075  df-fin 7076  df-oi 7439  df-har 7486  df-card 7786  df-aleph 7787
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