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Theorem cardsdomelir 8406
Description: A cardinal strictly dominates its members. Equivalent to Proposition 10.37 of [TakeutiZaring] p. 93. This is half of the assertion cardsdomel 8407 and can be proven without the AC. (Contributed by Mario Carneiro, 15-Jan-2013.)
Assertion
Ref Expression
cardsdomelir  |-  ( A  e.  ( card `  B
)  ->  A  ~<  B )

Proof of Theorem cardsdomelir
StepHypRef Expression
1 cardon 8377 . . . 4  |-  ( card `  B )  e.  On
21onelssi 5550 . . . 4  |-  ( A  e.  ( card `  B
)  ->  A  C_  ( card `  B ) )
3 ssdomg 7622 . . . 4  |-  ( (
card `  B )  e.  On  ->  ( A  C_  ( card `  B
)  ->  A  ~<_  ( card `  B ) ) )
41, 2, 3mpsyl 65 . . 3  |-  ( A  e.  ( card `  B
)  ->  A  ~<_  ( card `  B ) )
5 elfvdm 5907 . . . 4  |-  ( A  e.  ( card `  B
)  ->  B  e.  dom  card )
6 cardid2 8386 . . . 4  |-  ( B  e.  dom  card  ->  (
card `  B )  ~~  B )
75, 6syl 17 . . 3  |-  ( A  e.  ( card `  B
)  ->  ( card `  B )  ~~  B
)
8 domentr 7635 . . 3  |-  ( ( A  ~<_  ( card `  B
)  /\  ( card `  B )  ~~  B
)  ->  A  ~<_  B )
94, 7, 8syl2anc 665 . 2  |-  ( A  e.  ( card `  B
)  ->  A  ~<_  B )
10 cardne 8398 . 2  |-  ( A  e.  ( card `  B
)  ->  -.  A  ~~  B )
11 brsdom 7599 . 2  |-  ( A 
~<  B  <->  ( A  ~<_  B  /\  -.  A  ~~  B ) )
129, 10, 11sylanbrc 668 1  |-  ( A  e.  ( card `  B
)  ->  A  ~<  B )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    e. wcel 1870    C_ wss 3442   class class class wbr 4426   dom cdm 4854   Oncon0 5442   ` cfv 5601    ~~ cen 7574    ~<_ cdom 7575    ~< csdm 7576   cardccrd 8368
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 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
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 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-ord 5445  df-on 5446  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-en 7578  df-dom 7579  df-sdom 7580  df-card 8372
This theorem is referenced by:  cardsdomel  8407  pwsdompw  8632  alephval2  8995  pwcfsdom  9006  tskcard  9205
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