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Theorem cardsdomelir 7852
Description: A cardinal strictly dominates its members. Equivalent to Proposition 10.37 of [TakeutiZaring] p. 93. This is half of the assertion cardsdomel 7853 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 7823 . . . 4  |-  ( card `  B )  e.  On
21onelssi 4682 . . . 4  |-  ( A  e.  ( card `  B
)  ->  A  C_  ( card `  B ) )
3 ssdomg 7145 . . . 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 5749 . . . 4  |-  ( A  e.  ( card `  B
)  ->  B  e.  dom  card )
6 cardid2 7832 . . . 4  |-  ( B  e.  dom  card  ->  (
card `  B )  ~~  B )
75, 6syl 16 . . 3  |-  ( A  e.  ( card `  B
)  ->  ( card `  B )  ~~  B
)
8 domentr 7158 . . 3  |-  ( ( A  ~<_  ( card `  B
)  /\  ( card `  B )  ~~  B
)  ->  A  ~<_  B )
94, 7, 8syl2anc 643 . 2  |-  ( A  e.  ( card `  B
)  ->  A  ~<_  B )
10 cardne 7844 . 2  |-  ( A  e.  ( card `  B
)  ->  -.  A  ~~  B )
11 brsdom 7122 . 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 1725    C_ wss 3312   class class class wbr 4204   Oncon0 4573   dom cdm 4870   ` cfv 5446    ~~ cen 7098    ~<_ cdom 7099    ~< csdm 7100   cardccrd 7814
This theorem is referenced by:  cardsdomel  7853  pwsdompw  8076  alephval2  8439  pwcfsdom  8450  tskcard  8648
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-en 7102  df-dom 7103  df-sdom 7104  df-card 7818
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