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Theorem alephdom2 7861
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 7860 . . . 4  |-  ( ( B  e.  On  /\  A  e.  On )  ->  ( B  e.  (
aleph `  A )  <->  B  ~<  (
aleph `  A ) ) )
21ancoms 439 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( B  e.  (
aleph `  A )  <->  B  ~<  (
aleph `  A ) ) )
32notbid 285 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( -.  B  e.  ( aleph `  A )  <->  -.  B  ~<  ( aleph `  A ) ) )
4 alephon 7843 . . . . 5  |-  ( aleph `  A )  e.  On
54onordi 4600 . . . 4  |-  Ord  ( aleph `  A )
6 eloni 4505 . . . 4  |-  ( B  e.  On  ->  Ord  B )
7 ordtri1 4528 . . . 4  |-  ( ( Ord  ( aleph `  A
)  /\  Ord  B )  ->  ( ( aleph `  A )  C_  B  <->  -.  B  e.  ( aleph `  A ) ) )
85, 6, 7sylancr 644 . . 3  |-  ( B  e.  On  ->  (
( aleph `  A )  C_  B  <->  -.  B  e.  ( aleph `  A )
) )
98adantl 452 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( aleph `  A
)  C_  B  <->  -.  B  e.  ( aleph `  A )
) )
10 domtriord 7150 . . . 4  |-  ( ( ( aleph `  A )  e.  On  /\  B  e.  On )  ->  (
( aleph `  A )  ~<_  B 
<->  -.  B  ~<  ( aleph `  A ) ) )
114, 10mpan 651 . . 3  |-  ( B  e.  On  ->  (
( aleph `  A )  ~<_  B 
<->  -.  B  ~<  ( aleph `  A ) ) )
1211adantl 452 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( aleph `  A
)  ~<_  B  <->  -.  B  ~<  ( aleph `  A )
) )
133, 9, 123bitr4d 276 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 176    /\ wa 358    e. wcel 1715    C_ wss 3238   class class class wbr 4125   Ord word 4494   Oncon0 4495   ` cfv 5358    ~<_ cdom 7004    ~< csdm 7005   alephcale 7716
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-inf2 7489
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-int 3965  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-se 4456  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-isom 5367  df-riota 6446  df-recs 6530  df-rdg 6565  df-er 6802  df-en 7007  df-dom 7008  df-sdom 7009  df-fin 7010  df-oi 7372  df-har 7419  df-card 7719  df-aleph 7720
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