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Theorem carddomi2 8430
Description: Two sets have the dominance relationship if their cardinalities have the subset relationship and one is numerable. See also carddom 9005, which uses AC. (Contributed by Mario Carneiro, 11-Jan-2013.) (Revised by Mario Carneiro, 29-Apr-2015.)
Assertion
Ref Expression
carddomi2  |-  ( ( A  e.  dom  card  /\  B  e.  V )  ->  ( ( card `  A )  C_  ( card `  B )  ->  A  ~<_  B ) )

Proof of Theorem carddomi2
StepHypRef Expression
1 cardnueq0 8424 . . . . . 6  |-  ( A  e.  dom  card  ->  ( ( card `  A
)  =  (/)  <->  A  =  (/) ) )
21adantr 471 . . . . 5  |-  ( ( A  e.  dom  card  /\  B  e.  V )  ->  ( ( card `  A )  =  (/)  <->  A  =  (/) ) )
32biimpa 491 . . . 4  |-  ( ( ( A  e.  dom  card  /\  B  e.  V
)  /\  ( card `  A )  =  (/) )  ->  A  =  (/) )
4 0domg 7725 . . . . 5  |-  ( B  e.  V  ->  (/)  ~<_  B )
54ad2antlr 738 . . . 4  |-  ( ( ( A  e.  dom  card  /\  B  e.  V
)  /\  ( card `  A )  =  (/) )  ->  (/)  ~<_  B )
63, 5eqbrtrd 4437 . . 3  |-  ( ( ( A  e.  dom  card  /\  B  e.  V
)  /\  ( card `  A )  =  (/) )  ->  A  ~<_  B )
76a1d 26 . 2  |-  ( ( ( A  e.  dom  card  /\  B  e.  V
)  /\  ( card `  A )  =  (/) )  ->  ( ( card `  A )  C_  ( card `  B )  ->  A  ~<_  B ) )
8 fvex 5898 . . . . 5  |-  ( card `  B )  e.  _V
9 simprr 771 . . . . 5  |-  ( ( ( A  e.  dom  card  /\  B  e.  V
)  /\  ( ( card `  A )  =/=  (/)  /\  ( card `  A
)  C_  ( card `  B ) ) )  ->  ( card `  A
)  C_  ( card `  B ) )
10 ssdomg 7641 . . . . 5  |-  ( (
card `  B )  e.  _V  ->  ( ( card `  A )  C_  ( card `  B )  ->  ( card `  A
)  ~<_  ( card `  B
) ) )
118, 9, 10mpsyl 65 . . . 4  |-  ( ( ( A  e.  dom  card  /\  B  e.  V
)  /\  ( ( card `  A )  =/=  (/)  /\  ( card `  A
)  C_  ( card `  B ) ) )  ->  ( card `  A
)  ~<_  ( card `  B
) )
12 cardid2 8413 . . . . . 6  |-  ( A  e.  dom  card  ->  (
card `  A )  ~~  A )
1312ad2antrr 737 . . . . 5  |-  ( ( ( A  e.  dom  card  /\  B  e.  V
)  /\  ( ( card `  A )  =/=  (/)  /\  ( card `  A
)  C_  ( card `  B ) ) )  ->  ( card `  A
)  ~~  A )
14 simprl 769 . . . . . . 7  |-  ( ( ( A  e.  dom  card  /\  B  e.  V
)  /\  ( ( card `  A )  =/=  (/)  /\  ( card `  A
)  C_  ( card `  B ) ) )  ->  ( card `  A
)  =/=  (/) )
15 ssn0 3779 . . . . . . 7  |-  ( ( ( card `  A
)  C_  ( card `  B )  /\  ( card `  A )  =/=  (/) )  ->  ( card `  B )  =/=  (/) )
169, 14, 15syl2anc 671 . . . . . 6  |-  ( ( ( A  e.  dom  card  /\  B  e.  V
)  /\  ( ( card `  A )  =/=  (/)  /\  ( card `  A
)  C_  ( card `  B ) ) )  ->  ( card `  B
)  =/=  (/) )
17 ndmfv 5912 . . . . . . 7  |-  ( -.  B  e.  dom  card  -> 
( card `  B )  =  (/) )
1817necon1ai 2663 . . . . . 6  |-  ( (
card `  B )  =/=  (/)  ->  B  e.  dom  card )
19 cardid2 8413 . . . . . 6  |-  ( B  e.  dom  card  ->  (
card `  B )  ~~  B )
2016, 18, 193syl 18 . . . . 5  |-  ( ( ( A  e.  dom  card  /\  B  e.  V
)  /\  ( ( card `  A )  =/=  (/)  /\  ( card `  A
)  C_  ( card `  B ) ) )  ->  ( card `  B
)  ~~  B )
21 domen1 7740 . . . . . 6  |-  ( (
card `  A )  ~~  A  ->  ( (
card `  A )  ~<_  ( card `  B )  <->  A  ~<_  ( card `  B
) ) )
22 domen2 7741 . . . . . 6  |-  ( (
card `  B )  ~~  B  ->  ( A  ~<_  ( card `  B
)  <->  A  ~<_  B )
)
2321, 22sylan9bb 711 . . . . 5  |-  ( ( ( card `  A
)  ~~  A  /\  ( card `  B )  ~~  B )  ->  (
( card `  A )  ~<_  ( card `  B )  <->  A  ~<_  B ) )
2413, 20, 23syl2anc 671 . . . 4  |-  ( ( ( A  e.  dom  card  /\  B  e.  V
)  /\  ( ( card `  A )  =/=  (/)  /\  ( card `  A
)  C_  ( card `  B ) ) )  ->  ( ( card `  A )  ~<_  ( card `  B )  <->  A  ~<_  B ) )
2511, 24mpbid 215 . . 3  |-  ( ( ( A  e.  dom  card  /\  B  e.  V
)  /\  ( ( card `  A )  =/=  (/)  /\  ( card `  A
)  C_  ( card `  B ) ) )  ->  A  ~<_  B )
2625expr 624 . 2  |-  ( ( ( A  e.  dom  card  /\  B  e.  V
)  /\  ( card `  A )  =/=  (/) )  -> 
( ( card `  A
)  C_  ( card `  B )  ->  A  ~<_  B ) )
277, 26pm2.61dane 2723 1  |-  ( ( A  e.  dom  card  /\  B  e.  V )  ->  ( ( card `  A )  C_  ( card `  B )  ->  A  ~<_  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 375    = wceq 1455    e. wcel 1898    =/= wne 2633   _Vcvv 3057    C_ wss 3416   (/)c0 3743   class class class wbr 4416   dom cdm 4853   ` cfv 5601    ~~ cen 7592    ~<_ cdom 7593   cardccrd 8395
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6610
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-rab 2758  df-v 3059  df-sbc 3280  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-int 4249  df-br 4417  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-ord 5445  df-on 5446  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-er 7389  df-en 7596  df-dom 7597  df-card 8399
This theorem is referenced by:  carddom2  8437
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