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Theorem cardsdomelir 7793
Description: A cardinal strictly dominates its members. Equivalent to Proposition 10.37 of [TakeutiZaring] p. 93. This is half of the assertion cardsdomel 7794 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 7764 . . . 4  |-  ( card `  B )  e.  On
21onelssi 4630 . . . 4  |-  ( A  e.  ( card `  B
)  ->  A  C_  ( card `  B ) )
3 ssdomg 7089 . . . 4  |-  ( (
card `  B )  e.  On  ->  ( A  C_  ( card `  B
)  ->  A  ~<_  ( card `  B ) ) )
41, 2, 3mpsyl 61 . . 3  |-  ( A  e.  ( card `  B
)  ->  A  ~<_  ( card `  B ) )
5 elfvdm 5697 . . . 4  |-  ( A  e.  ( card `  B
)  ->  B  e.  dom  card )
6 cardid2 7773 . . . 4  |-  ( B  e.  dom  card  ->  (
card `  B )  ~~  B )
75, 6syl 16 . . 3  |-  ( A  e.  ( card `  B
)  ->  ( card `  B )  ~~  B
)
8 domentr 7102 . . 3  |-  ( ( A  ~<_  ( card `  B
)  /\  ( card `  B )  ~~  B
)  ->  A  ~<_  B )
94, 7, 8syl2anc 643 . 2  |-  ( A  e.  ( card `  B
)  ->  A  ~<_  B )
10 cardne 7785 . 2  |-  ( A  e.  ( card `  B
)  ->  -.  A  ~~  B )
11 brsdom 7066 . 2  |-  ( A 
~<  B  <->  ( A  ~<_  B  /\  -.  A  ~~  B ) )
129, 10, 11sylanbrc 646 1  |-  ( A  e.  ( card `  B
)  ->  A  ~<  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    e. wcel 1717    C_ wss 3263   class class class wbr 4153   Oncon0 4522   dom cdm 4818   ` cfv 5394    ~~ cen 7042    ~<_ cdom 7043    ~< csdm 7044   cardccrd 7755
This theorem is referenced by:  cardsdomel  7794  pwsdompw  8017  alephval2  8380  pwcfsdom  8391  tskcard  8589
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-en 7046  df-dom 7047  df-sdom 7048  df-card 7759
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