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Theorem cardsdom2 7867
Description: A numerable set is strictly dominated by another iff their cardinalities are strictly ordered. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Revised by Mario Carneiro, 29-Apr-2015.)
Assertion
Ref Expression
cardsdom2  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card )  ->  ( ( card `  A )  e.  (
card `  B )  <->  A 
~<  B ) )

Proof of Theorem cardsdom2
StepHypRef Expression
1 carddom2 7856 . . 3  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card )  ->  ( ( card `  A )  C_  ( card `  B )  <->  A  ~<_  B ) )
2 carden2 7866 . . . 4  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card )  ->  ( ( card `  A )  =  (
card `  B )  <->  A 
~~  B ) )
32necon3abid 2631 . . 3  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card )  ->  ( ( card `  A )  =/=  ( card `  B )  <->  -.  A  ~~  B ) )
41, 3anbi12d 692 . 2  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card )  ->  ( ( (
card `  A )  C_  ( card `  B
)  /\  ( card `  A )  =/=  ( card `  B ) )  <-> 
( A  ~<_  B  /\  -.  A  ~~  B ) ) )
5 cardon 7823 . . 3  |-  ( card `  A )  e.  On
6 cardon 7823 . . 3  |-  ( card `  B )  e.  On
7 onelpss 4613 . . 3  |-  ( ( ( card `  A
)  e.  On  /\  ( card `  B )  e.  On )  ->  (
( card `  A )  e.  ( card `  B
)  <->  ( ( card `  A )  C_  ( card `  B )  /\  ( card `  A )  =/=  ( card `  B
) ) ) )
85, 6, 7mp2an 654 . 2  |-  ( (
card `  A )  e.  ( card `  B
)  <->  ( ( card `  A )  C_  ( card `  B )  /\  ( card `  A )  =/=  ( card `  B
) ) )
9 brsdom 7122 . 2  |-  ( A 
~<  B  <->  ( A  ~<_  B  /\  -.  A  ~~  B ) )
104, 8, 93bitr4g 280 1  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card )  ->  ( ( card `  A )  e.  (
card `  B )  <->  A 
~<  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    e. wcel 1725    =/= wne 2598    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:  domtri2  7868  nnsdomel  7869  indcardi  7914  sdom2en01  8174  cardsdom  8422  smobeth  8453  hargch  8544
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-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-card 7818
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