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Theorem alephdom2 8372
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 8371 . . . 4  |-  ( ( B  e.  On  /\  A  e.  On )  ->  ( B  e.  (
aleph `  A )  <->  B  ~<  (
aleph `  A ) ) )
21ancoms 453 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( B  e.  (
aleph `  A )  <->  B  ~<  (
aleph `  A ) ) )
32notbid 294 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( -.  B  e.  ( aleph `  A )  <->  -.  B  ~<  ( aleph `  A ) ) )
4 alephon 8354 . . . . 5  |-  ( aleph `  A )  e.  On
54onordi 4934 . . . 4  |-  Ord  ( aleph `  A )
6 eloni 4840 . . . 4  |-  ( B  e.  On  ->  Ord  B )
7 ordtri1 4863 . . . 4  |-  ( ( Ord  ( aleph `  A
)  /\  Ord  B )  ->  ( ( aleph `  A )  C_  B  <->  -.  B  e.  ( aleph `  A ) ) )
85, 6, 7sylancr 663 . . 3  |-  ( B  e.  On  ->  (
( aleph `  A )  C_  B  <->  -.  B  e.  ( aleph `  A )
) )
98adantl 466 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( aleph `  A
)  C_  B  <->  -.  B  e.  ( aleph `  A )
) )
10 domtriord 7570 . . . 4  |-  ( ( ( aleph `  A )  e.  On  /\  B  e.  On )  ->  (
( aleph `  A )  ~<_  B 
<->  -.  B  ~<  ( aleph `  A ) ) )
114, 10mpan 670 . . 3  |-  ( B  e.  On  ->  (
( aleph `  A )  ~<_  B 
<->  -.  B  ~<  ( aleph `  A ) ) )
1211adantl 466 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( aleph `  A
)  ~<_  B  <->  -.  B  ~<  ( aleph `  A )
) )
133, 9, 123bitr4d 285 1  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( aleph `  A
)  C_  B  <->  ( aleph `  A )  ~<_  B ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    e. wcel 1758    C_ wss 3439   class class class wbr 4403   Ord word 4829   Oncon0 4830   ` cfv 5529    ~<_ cdom 7421    ~< csdm 7422   alephcale 8221
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-inf2 7962
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-se 4791  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-isom 5538  df-riota 6164  df-om 6590  df-recs 6945  df-rdg 6979  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-oi 7839  df-har 7888  df-card 8224  df-aleph 8225
This theorem is referenced by: (None)
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