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Theorem carddomi2 8243
Description: Two sets have the dominance relationship if their cardinalities have the subset relationship and one is numerable. See also carddom 8821, 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 8237 . . . . . 6  |-  ( A  e.  dom  card  ->  ( ( card `  A
)  =  (/)  <->  A  =  (/) ) )
21adantr 465 . . . . 5  |-  ( ( A  e.  dom  card  /\  B  e.  V )  ->  ( ( card `  A )  =  (/)  <->  A  =  (/) ) )
32biimpa 484 . . . 4  |-  ( ( ( A  e.  dom  card  /\  B  e.  V
)  /\  ( card `  A )  =  (/) )  ->  A  =  (/) )
4 0domg 7540 . . . . 5  |-  ( B  e.  V  ->  (/)  ~<_  B )
54ad2antlr 726 . . . 4  |-  ( ( ( A  e.  dom  card  /\  B  e.  V
)  /\  ( card `  A )  =  (/) )  ->  (/)  ~<_  B )
63, 5eqbrtrd 4412 . . 3  |-  ( ( ( A  e.  dom  card  /\  B  e.  V
)  /\  ( card `  A )  =  (/) )  ->  A  ~<_  B )
76a1d 25 . 2  |-  ( ( ( A  e.  dom  card  /\  B  e.  V
)  /\  ( card `  A )  =  (/) )  ->  ( ( card `  A )  C_  ( card `  B )  ->  A  ~<_  B ) )
8 fvex 5801 . . . . 5  |-  ( card `  B )  e.  _V
9 simprr 756 . . . . 5  |-  ( ( ( A  e.  dom  card  /\  B  e.  V
)  /\  ( ( card `  A )  =/=  (/)  /\  ( card `  A
)  C_  ( card `  B ) ) )  ->  ( card `  A
)  C_  ( card `  B ) )
10 ssdomg 7457 . . . . 5  |-  ( (
card `  B )  e.  _V  ->  ( ( card `  A )  C_  ( card `  B )  ->  ( card `  A
)  ~<_  ( card `  B
) ) )
118, 9, 10mpsyl 63 . . . 4  |-  ( ( ( A  e.  dom  card  /\  B  e.  V
)  /\  ( ( card `  A )  =/=  (/)  /\  ( card `  A
)  C_  ( card `  B ) ) )  ->  ( card `  A
)  ~<_  ( card `  B
) )
12 cardid2 8226 . . . . . 6  |-  ( A  e.  dom  card  ->  (
card `  A )  ~~  A )
1312ad2antrr 725 . . . . 5  |-  ( ( ( A  e.  dom  card  /\  B  e.  V
)  /\  ( ( card `  A )  =/=  (/)  /\  ( card `  A
)  C_  ( card `  B ) ) )  ->  ( card `  A
)  ~~  A )
14 simprl 755 . . . . . . 7  |-  ( ( ( A  e.  dom  card  /\  B  e.  V
)  /\  ( ( card `  A )  =/=  (/)  /\  ( card `  A
)  C_  ( card `  B ) ) )  ->  ( card `  A
)  =/=  (/) )
15 ssn0 3770 . . . . . . 7  |-  ( ( ( card `  A
)  C_  ( card `  B )  /\  ( card `  A )  =/=  (/) )  ->  ( card `  B )  =/=  (/) )
169, 14, 15syl2anc 661 . . . . . 6  |-  ( ( ( A  e.  dom  card  /\  B  e.  V
)  /\  ( ( card `  A )  =/=  (/)  /\  ( card `  A
)  C_  ( card `  B ) ) )  ->  ( card `  B
)  =/=  (/) )
17 ndmfv 5815 . . . . . . 7  |-  ( -.  B  e.  dom  card  -> 
( card `  B )  =  (/) )
1817necon1ai 2679 . . . . . 6  |-  ( (
card `  B )  =/=  (/)  ->  B  e.  dom  card )
19 cardid2 8226 . . . . . 6  |-  ( B  e.  dom  card  ->  (
card `  B )  ~~  B )
2016, 18, 193syl 20 . . . . 5  |-  ( ( ( A  e.  dom  card  /\  B  e.  V
)  /\  ( ( card `  A )  =/=  (/)  /\  ( card `  A
)  C_  ( card `  B ) ) )  ->  ( card `  B
)  ~~  B )
21 domen1 7555 . . . . . 6  |-  ( (
card `  A )  ~~  A  ->  ( (
card `  A )  ~<_  ( card `  B )  <->  A  ~<_  ( card `  B
) ) )
22 domen2 7556 . . . . . 6  |-  ( (
card `  B )  ~~  B  ->  ( A  ~<_  ( card `  B
)  <->  A  ~<_  B )
)
2321, 22sylan9bb 699 . . . . 5  |-  ( ( ( card `  A
)  ~~  A  /\  ( card `  B )  ~~  B )  ->  (
( card `  A )  ~<_  ( card `  B )  <->  A  ~<_  B ) )
2413, 20, 23syl2anc 661 . . . 4  |-  ( ( ( A  e.  dom  card  /\  B  e.  V
)  /\  ( ( card `  A )  =/=  (/)  /\  ( card `  A
)  C_  ( card `  B ) ) )  ->  ( ( card `  A )  ~<_  ( card `  B )  <->  A  ~<_  B ) )
2511, 24mpbid 210 . . 3  |-  ( ( ( A  e.  dom  card  /\  B  e.  V
)  /\  ( ( card `  A )  =/=  (/)  /\  ( card `  A
)  C_  ( card `  B ) ) )  ->  A  ~<_  B )
2625expr 615 . 2  |-  ( ( ( A  e.  dom  card  /\  B  e.  V
)  /\  ( card `  A )  =/=  (/) )  -> 
( ( card `  A
)  C_  ( card `  B )  ->  A  ~<_  B ) )
277, 26pm2.61dane 2766 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 184    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2644   _Vcvv 3070    C_ wss 3428   (/)c0 3737   class class class wbr 4392   dom cdm 4940   ` cfv 5518    ~~ cen 7409    ~<_ cdom 7410   cardccrd 8208
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3072  df-sbc 3287  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-int 4229  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-eprel 4732  df-id 4736  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-er 7203  df-en 7413  df-dom 7414  df-card 8212
This theorem is referenced by:  carddom2  8250
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