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Theorem cardsdomel 7850
Description: A cardinal strictly dominates its members. Equivalent to Proposition 10.37 of [TakeutiZaring] p. 93. (Contributed by Mario Carneiro, 15-Jan-2013.) (Revised by Mario Carneiro, 4-Jun-2015.)
Assertion
Ref Expression
cardsdomel  |-  ( ( A  e.  On  /\  B  e.  dom  card )  ->  ( A  ~<  B  <->  A  e.  ( card `  B )
) )

Proof of Theorem cardsdomel
StepHypRef Expression
1 cardid2 7829 . . . . . . 7  |-  ( B  e.  dom  card  ->  (
card `  B )  ~~  B )
21ensymd 7149 . . . . . 6  |-  ( B  e.  dom  card  ->  B 
~~  ( card `  B
) )
3 sdomentr 7232 . . . . . 6  |-  ( ( A  ~<  B  /\  B  ~~  ( card `  B
) )  ->  A  ~<  ( card `  B
) )
42, 3sylan2 461 . . . . 5  |-  ( ( A  ~<  B  /\  B  e.  dom  card )  ->  A  ~<  ( card `  B ) )
5 ssdomg 7144 . . . . . . . 8  |-  ( A  e.  On  ->  (
( card `  B )  C_  A  ->  ( card `  B )  ~<_  A ) )
6 cardon 7820 . . . . . . . . 9  |-  ( card `  B )  e.  On
7 domtriord 7244 . . . . . . . . 9  |-  ( ( ( card `  B
)  e.  On  /\  A  e.  On )  ->  ( ( card `  B
)  ~<_  A  <->  -.  A  ~<  ( card `  B
) ) )
86, 7mpan 652 . . . . . . . 8  |-  ( A  e.  On  ->  (
( card `  B )  ~<_  A 
<->  -.  A  ~<  ( card `  B ) ) )
95, 8sylibd 206 . . . . . . 7  |-  ( A  e.  On  ->  (
( card `  B )  C_  A  ->  -.  A  ~<  ( card `  B
) ) )
109con2d 109 . . . . . 6  |-  ( A  e.  On  ->  ( A  ~<  ( card `  B
)  ->  -.  ( card `  B )  C_  A ) )
11 ontri1 4607 . . . . . . . 8  |-  ( ( ( card `  B
)  e.  On  /\  A  e.  On )  ->  ( ( card `  B
)  C_  A  <->  -.  A  e.  ( card `  B
) ) )
126, 11mpan 652 . . . . . . 7  |-  ( A  e.  On  ->  (
( card `  B )  C_  A  <->  -.  A  e.  ( card `  B )
) )
1312con2bid 320 . . . . . 6  |-  ( A  e.  On  ->  ( A  e.  ( card `  B )  <->  -.  ( card `  B )  C_  A ) )
1410, 13sylibrd 226 . . . . 5  |-  ( A  e.  On  ->  ( A  ~<  ( card `  B
)  ->  A  e.  ( card `  B )
) )
154, 14syl5 30 . . . 4  |-  ( A  e.  On  ->  (
( A  ~<  B  /\  B  e.  dom  card )  ->  A  e.  ( card `  B ) ) )
1615exp3acom23 1381 . . 3  |-  ( A  e.  On  ->  ( B  e.  dom  card  ->  ( A  ~<  B  ->  A  e.  ( card `  B
) ) ) )
1716imp 419 . 2  |-  ( ( A  e.  On  /\  B  e.  dom  card )  ->  ( A  ~<  B  ->  A  e.  ( card `  B ) ) )
18 cardsdomelir 7849 . 2  |-  ( A  e.  ( card `  B
)  ->  A  ~<  B )
1917, 18impbid1 195 1  |-  ( ( A  e.  On  /\  B  e.  dom  card )  ->  ( A  ~<  B  <->  A  e.  ( card `  B )
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    e. wcel 1725    C_ wss 3312   class class class wbr 4204   Oncon0 4573   dom cdm 4869   ` cfv 5445    ~~ cen 7097    ~<_ cdom 7098    ~< csdm 7099   cardccrd 7811
This theorem is referenced by:  iscard  7851  cardval2  7867  infxpenlem  7884  alephnbtwn  7941  alephnbtwn2  7942  alephord2  7946  alephsdom  7956  pwsdompw  8073  inaprc  8700
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 4692
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 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-er 6896  df-en 7101  df-dom 7102  df-sdom 7103  df-card 7815
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