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Theorem cardsdom2 7621
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 7610 . . 3  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card )  ->  ( ( card `  A )  C_  ( card `  B )  <->  A  ~<_  B ) )
2 carden2 7620 . . . 4  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card )  ->  ( ( card `  A )  =  (
card `  B )  <->  A 
~~  B ) )
32necon3abid 2479 . . 3  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card )  ->  ( ( card `  A )  =/=  ( card `  B )  <->  -.  A  ~~  B ) )
41, 3anbi12d 691 . 2  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card )  ->  ( ( (
card `  A )  C_  ( card `  B
)  /\  ( card `  A )  =/=  ( card `  B ) )  <-> 
( A  ~<_  B  /\  -.  A  ~~  B ) ) )
5 cardon 7577 . . 3  |-  ( card `  A )  e.  On
6 cardon 7577 . . 3  |-  ( card `  B )  e.  On
7 onelpss 4432 . . 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 653 . 2  |-  ( (
card `  A )  e.  ( card `  B
)  <->  ( ( card `  A )  C_  ( card `  B )  /\  ( card `  A )  =/=  ( card `  B
) ) )
9 brsdom 6884 . 2  |-  ( A 
~<  B  <->  ( A  ~<_  B  /\  -.  A  ~~  B ) )
104, 8, 93bitr4g 279 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 176    /\ wa 358    e. wcel 1684    =/= wne 2446    C_ wss 3152   class class class wbr 4023   Oncon0 4392   dom cdm 4689   ` cfv 5255    ~~ cen 6860    ~<_ cdom 6861    ~< csdm 6862   cardccrd 7568
This theorem is referenced by:  domtri2  7622  nnsdomel  7623  indcardi  7668  sdom2en01  7928  cardsdom  8177  smobeth  8208  hargch  8299
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-card 7572
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