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Theorem cardsdom2 7808
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 7797 . . 3  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card )  ->  ( ( card `  A )  C_  ( card `  B )  <->  A  ~<_  B ) )
2 carden2 7807 . . . 4  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card )  ->  ( ( card `  A )  =  (
card `  B )  <->  A 
~~  B ) )
32necon3abid 2583 . . 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 7764 . . 3  |-  ( card `  A )  e.  On
6 cardon 7764 . . 3  |-  ( card `  B )  e.  On
7 onelpss 4562 . . 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 7066 . 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 1717    =/= wne 2550    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:  domtri2  7809  nnsdomel  7810  indcardi  7855  sdom2en01  8115  cardsdom  8363  smobeth  8394  hargch  8485
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-er 6841  df-en 7046  df-dom 7047  df-sdom 7048  df-card 7759
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