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Theorem cardsdomelir 7608
Description: A cardinal strictly dominates its members. Equivalent to Proposition 10.37 of [TakeutiZaring] p. 93. This is half of the assertion cardsdomel 7609 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 7579 . . . 4  |-  ( card `  B )  e.  On
21onelssi 4503 . . . 4  |-  ( A  e.  ( card `  B
)  ->  A  C_  ( card `  B ) )
3 ssdomg 6909 . . . 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 5556 . . . 4  |-  ( A  e.  ( card `  B
)  ->  B  e.  dom  card )
6 cardid2 7588 . . . 4  |-  ( B  e.  dom  card  ->  (
card `  B )  ~~  B )
75, 6syl 15 . . 3  |-  ( A  e.  ( card `  B
)  ->  ( card `  B )  ~~  B
)
8 domentr 6922 . . 3  |-  ( ( A  ~<_  ( card `  B
)  /\  ( card `  B )  ~~  B
)  ->  A  ~<_  B )
94, 7, 8syl2anc 642 . 2  |-  ( A  e.  ( card `  B
)  ->  A  ~<_  B )
10 cardne 7600 . 2  |-  ( A  e.  ( card `  B
)  ->  -.  A  ~~  B )
11 brsdom 6886 . 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 1686    C_ wss 3154   class class class wbr 4025   Oncon0 4394   dom cdm 4691   ` cfv 5257    ~~ cen 6862    ~<_ cdom 6863    ~< csdm 6864   cardccrd 7570
This theorem is referenced by:  cardsdomel  7609  pwsdompw  7832  alephval2  8196  pwcfsdom  8207  tskcard  8405
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514
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 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-sbc 2994  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-int 3865  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-we 4356  df-ord 4397  df-on 4398  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-en 6866  df-dom 6867  df-sdom 6868  df-card 7574
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