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Theorem carddomi2 8282
Description: Two sets have the dominance relationship if their cardinalities have the subset relationship and one is numerable. See also carddom 8860, 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 8276 . . . . . 6  |-  ( A  e.  dom  card  ->  ( ( card `  A
)  =  (/)  <->  A  =  (/) ) )
21adantr 463 . . . . 5  |-  ( ( A  e.  dom  card  /\  B  e.  V )  ->  ( ( card `  A )  =  (/)  <->  A  =  (/) ) )
32biimpa 482 . . . 4  |-  ( ( ( A  e.  dom  card  /\  B  e.  V
)  /\  ( card `  A )  =  (/) )  ->  A  =  (/) )
4 0domg 7581 . . . . 5  |-  ( B  e.  V  ->  (/)  ~<_  B )
54ad2antlr 724 . . . 4  |-  ( ( ( A  e.  dom  card  /\  B  e.  V
)  /\  ( card `  A )  =  (/) )  ->  (/)  ~<_  B )
63, 5eqbrtrd 4400 . . 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 5797 . . . . 5  |-  ( card `  B )  e.  _V
9 simprr 755 . . . . 5  |-  ( ( ( A  e.  dom  card  /\  B  e.  V
)  /\  ( ( card `  A )  =/=  (/)  /\  ( card `  A
)  C_  ( card `  B ) ) )  ->  ( card `  A
)  C_  ( card `  B ) )
10 ssdomg 7498 . . . . 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 8265 . . . . . 6  |-  ( A  e.  dom  card  ->  (
card `  A )  ~~  A )
1312ad2antrr 723 . . . . 5  |-  ( ( ( A  e.  dom  card  /\  B  e.  V
)  /\  ( ( card `  A )  =/=  (/)  /\  ( card `  A
)  C_  ( card `  B ) ) )  ->  ( card `  A
)  ~~  A )
14 simprl 754 . . . . . . 7  |-  ( ( ( A  e.  dom  card  /\  B  e.  V
)  /\  ( ( card `  A )  =/=  (/)  /\  ( card `  A
)  C_  ( card `  B ) ) )  ->  ( card `  A
)  =/=  (/) )
15 ssn0 3758 . . . . . . 7  |-  ( ( ( card `  A
)  C_  ( card `  B )  /\  ( card `  A )  =/=  (/) )  ->  ( card `  B )  =/=  (/) )
169, 14, 15syl2anc 659 . . . . . 6  |-  ( ( ( A  e.  dom  card  /\  B  e.  V
)  /\  ( ( card `  A )  =/=  (/)  /\  ( card `  A
)  C_  ( card `  B ) ) )  ->  ( card `  B
)  =/=  (/) )
17 ndmfv 5811 . . . . . . 7  |-  ( -.  B  e.  dom  card  -> 
( card `  B )  =  (/) )
1817necon1ai 2623 . . . . . 6  |-  ( (
card `  B )  =/=  (/)  ->  B  e.  dom  card )
19 cardid2 8265 . . . . . 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 7596 . . . . . 6  |-  ( (
card `  A )  ~~  A  ->  ( (
card `  A )  ~<_  ( card `  B )  <->  A  ~<_  ( card `  B
) ) )
22 domen2 7597 . . . . . 6  |-  ( (
card `  B )  ~~  B  ->  ( A  ~<_  ( card `  B
)  <->  A  ~<_  B )
)
2321, 22sylan9bb 697 . . . . 5  |-  ( ( ( card `  A
)  ~~  A  /\  ( card `  B )  ~~  B )  ->  (
( card `  A )  ~<_  ( card `  B )  <->  A  ~<_  B ) )
2413, 20, 23syl2anc 659 . . . 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 613 . 2  |-  ( ( ( A  e.  dom  card  /\  B  e.  V
)  /\  ( card `  A )  =/=  (/) )  -> 
( ( card `  A
)  C_  ( card `  B )  ->  A  ~<_  B ) )
277, 26pm2.61dane 2710 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 367    = wceq 1399    e. wcel 1836    =/= wne 2587   _Vcvv 3047    C_ wss 3402   (/)c0 3724   class class class wbr 4380   dom cdm 4926   ` cfv 5509    ~~ cen 7450    ~<_ cdom 7451   cardccrd 8247
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-8 1838  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2016  ax-ext 2370  ax-sep 4501  ax-nul 4509  ax-pow 4556  ax-pr 4614  ax-un 6509
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2232  df-mo 2233  df-clab 2378  df-cleq 2384  df-clel 2387  df-nfc 2542  df-ne 2589  df-ral 2747  df-rex 2748  df-rab 2751  df-v 3049  df-sbc 3266  df-dif 3405  df-un 3407  df-in 3409  df-ss 3416  df-pss 3418  df-nul 3725  df-if 3871  df-pw 3942  df-sn 3958  df-pr 3960  df-tp 3962  df-op 3964  df-uni 4177  df-int 4213  df-br 4381  df-opab 4439  df-mpt 4440  df-tr 4474  df-eprel 4718  df-id 4722  df-po 4727  df-so 4728  df-fr 4765  df-we 4767  df-ord 4808  df-on 4809  df-xp 4932  df-rel 4933  df-cnv 4934  df-co 4935  df-dm 4936  df-rn 4937  df-res 4938  df-ima 4939  df-iota 5473  df-fun 5511  df-fn 5512  df-f 5513  df-f1 5514  df-fo 5515  df-f1o 5516  df-fv 5517  df-er 7247  df-en 7454  df-dom 7455  df-card 8251
This theorem is referenced by:  carddom2  8289
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