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Theorem cardsdomelir 7622
Description: A cardinal strictly dominates its members. Equivalent to Proposition 10.37 of [TakeutiZaring] p. 93. This is half of the assertion cardsdomel 7623 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 7593 . . . 4  |-  ( card `  B )  e.  On
21onelssi 4517 . . . 4  |-  ( A  e.  ( card `  B
)  ->  A  C_  ( card `  B ) )
3 ssdomg 6923 . . . 4  |-  ( (
card `  B )  e.  On  ->  ( A  C_  ( card `  B
)  ->  A  ~<_  ( card `  B ) ) )
41, 2, 3mpsyl 59 . . 3  |-  ( A  e.  ( card `  B
)  ->  A  ~<_  ( card `  B ) )
5 elfvdm 5570 . . . 4  |-  ( A  e.  ( card `  B
)  ->  B  e.  dom  card )
6 cardid2 7602 . . . 4  |-  ( B  e.  dom  card  ->  (
card `  B )  ~~  B )
75, 6syl 15 . . 3  |-  ( A  e.  ( card `  B
)  ->  ( card `  B )  ~~  B
)
8 domentr 6936 . . 3  |-  ( ( A  ~<_  ( card `  B
)  /\  ( card `  B )  ~~  B
)  ->  A  ~<_  B )
94, 7, 8syl2anc 642 . 2  |-  ( A  e.  ( card `  B
)  ->  A  ~<_  B )
10 cardne 7614 . 2  |-  ( A  e.  ( card `  B
)  ->  -.  A  ~~  B )
11 brsdom 6900 . 2  |-  ( A 
~<  B  <->  ( A  ~<_  B  /\  -.  A  ~~  B ) )
129, 10, 11sylanbrc 645 1  |-  ( A  e.  ( card `  B
)  ->  A  ~<  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    e. wcel 1696    C_ wss 3165   class class class wbr 4039   Oncon0 4408   dom cdm 4705   ` cfv 5271    ~~ cen 6876    ~<_ cdom 6877    ~< csdm 6878   cardccrd 7584
This theorem is referenced by:  cardsdomel  7623  pwsdompw  7846  alephval2  8210  pwcfsdom  8221  tskcard  8419
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-en 6880  df-dom 6881  df-sdom 6882  df-card 7588
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