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Theorem carddomi2 7790
Description: Two sets have the dominance relationship if their cardinalities have the subset relationship and one is numerable. See also carddom 8362, 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 7784 . . . . . 6  |-  ( A  e.  dom  card  ->  ( ( card `  A
)  =  (/)  <->  A  =  (/) ) )
21adantr 452 . . . . 5  |-  ( ( A  e.  dom  card  /\  B  e.  V )  ->  ( ( card `  A )  =  (/)  <->  A  =  (/) ) )
32biimpa 471 . . . 4  |-  ( ( ( A  e.  dom  card  /\  B  e.  V
)  /\  ( card `  A )  =  (/) )  ->  A  =  (/) )
4 0domg 7170 . . . . 5  |-  ( B  e.  V  ->  (/)  ~<_  B )
54ad2antlr 708 . . . 4  |-  ( ( ( A  e.  dom  card  /\  B  e.  V
)  /\  ( card `  A )  =  (/) )  ->  (/)  ~<_  B )
63, 5eqbrtrd 4173 . . 3  |-  ( ( ( A  e.  dom  card  /\  B  e.  V
)  /\  ( card `  A )  =  (/) )  ->  A  ~<_  B )
76a1d 23 . 2  |-  ( ( ( A  e.  dom  card  /\  B  e.  V
)  /\  ( card `  A )  =  (/) )  ->  ( ( card `  A )  C_  ( card `  B )  ->  A  ~<_  B ) )
8 fvex 5682 . . . . 5  |-  ( card `  B )  e.  _V
9 simprr 734 . . . . 5  |-  ( ( ( A  e.  dom  card  /\  B  e.  V
)  /\  ( ( card `  A )  =/=  (/)  /\  ( card `  A
)  C_  ( card `  B ) ) )  ->  ( card `  A
)  C_  ( card `  B ) )
10 ssdomg 7089 . . . . 5  |-  ( (
card `  B )  e.  _V  ->  ( ( card `  A )  C_  ( card `  B )  ->  ( card `  A
)  ~<_  ( card `  B
) ) )
118, 9, 10mpsyl 61 . . . 4  |-  ( ( ( A  e.  dom  card  /\  B  e.  V
)  /\  ( ( card `  A )  =/=  (/)  /\  ( card `  A
)  C_  ( card `  B ) ) )  ->  ( card `  A
)  ~<_  ( card `  B
) )
12 cardid2 7773 . . . . . 6  |-  ( A  e.  dom  card  ->  (
card `  A )  ~~  A )
1312ad2antrr 707 . . . . 5  |-  ( ( ( A  e.  dom  card  /\  B  e.  V
)  /\  ( ( card `  A )  =/=  (/)  /\  ( card `  A
)  C_  ( card `  B ) ) )  ->  ( card `  A
)  ~~  A )
14 simprl 733 . . . . . . 7  |-  ( ( ( A  e.  dom  card  /\  B  e.  V
)  /\  ( ( card `  A )  =/=  (/)  /\  ( card `  A
)  C_  ( card `  B ) ) )  ->  ( card `  A
)  =/=  (/) )
15 ssn0 3603 . . . . . . 7  |-  ( ( ( card `  A
)  C_  ( card `  B )  /\  ( card `  A )  =/=  (/) )  ->  ( card `  B )  =/=  (/) )
169, 14, 15syl2anc 643 . . . . . 6  |-  ( ( ( A  e.  dom  card  /\  B  e.  V
)  /\  ( ( card `  A )  =/=  (/)  /\  ( card `  A
)  C_  ( card `  B ) ) )  ->  ( card `  B
)  =/=  (/) )
17 ndmfv 5695 . . . . . . 7  |-  ( -.  B  e.  dom  card  -> 
( card `  B )  =  (/) )
1817necon1ai 2592 . . . . . 6  |-  ( (
card `  B )  =/=  (/)  ->  B  e.  dom  card )
19 cardid2 7773 . . . . . 6  |-  ( B  e.  dom  card  ->  (
card `  B )  ~~  B )
2016, 18, 193syl 19 . . . . 5  |-  ( ( ( A  e.  dom  card  /\  B  e.  V
)  /\  ( ( card `  A )  =/=  (/)  /\  ( card `  A
)  C_  ( card `  B ) ) )  ->  ( card `  B
)  ~~  B )
21 domen1 7185 . . . . . 6  |-  ( (
card `  A )  ~~  A  ->  ( (
card `  A )  ~<_  ( card `  B )  <->  A  ~<_  ( card `  B
) ) )
22 domen2 7186 . . . . . 6  |-  ( (
card `  B )  ~~  B  ->  ( A  ~<_  ( card `  B
)  <->  A  ~<_  B )
)
2321, 22sylan9bb 681 . . . . 5  |-  ( ( ( card `  A
)  ~~  A  /\  ( card `  B )  ~~  B )  ->  (
( card `  A )  ~<_  ( card `  B )  <->  A  ~<_  B ) )
2413, 20, 23syl2anc 643 . . . 4  |-  ( ( ( A  e.  dom  card  /\  B  e.  V
)  /\  ( ( card `  A )  =/=  (/)  /\  ( card `  A
)  C_  ( card `  B ) ) )  ->  ( ( card `  A )  ~<_  ( card `  B )  <->  A  ~<_  B ) )
2511, 24mpbid 202 . . 3  |-  ( ( ( A  e.  dom  card  /\  B  e.  V
)  /\  ( ( card `  A )  =/=  (/)  /\  ( card `  A
)  C_  ( card `  B ) ) )  ->  A  ~<_  B )
2625expr 599 . 2  |-  ( ( ( A  e.  dom  card  /\  B  e.  V
)  /\  ( card `  A )  =/=  (/) )  -> 
( ( card `  A
)  C_  ( card `  B )  ->  A  ~<_  B ) )
277, 26pm2.61dane 2628 1  |-  ( ( A  e.  dom  card  /\  B  e.  V )  ->  ( ( card `  A )  C_  ( card `  B )  ->  A  ~<_  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717    =/= wne 2550   _Vcvv 2899    C_ wss 3263   (/)c0 3571   class class class wbr 4153   dom cdm 4818   ` cfv 5394    ~~ cen 7042    ~<_ cdom 7043   cardccrd 7755
This theorem is referenced by:  carddom2  7797
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-card 7759
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