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Theorem carddomi2 7849
Description: Two sets have the dominance relationship if their cardinalities have the subset relationship and one is numerable. See also carddom 8421, 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 7843 . . . . . 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 7226 . . . . 5  |-  ( B  e.  V  ->  (/)  ~<_  B )
54ad2antlr 708 . . . 4  |-  ( ( ( A  e.  dom  card  /\  B  e.  V
)  /\  ( card `  A )  =  (/) )  ->  (/)  ~<_  B )
63, 5eqbrtrd 4224 . . 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 5734 . . . . 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 7145 . . . . 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 7832 . . . . . 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 3652 . . . . . . 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 5747 . . . . . . 7  |-  ( -.  B  e.  dom  card  -> 
( card `  B )  =  (/) )
1817necon1ai 2640 . . . . . 6  |-  ( (
card `  B )  =/=  (/)  ->  B  e.  dom  card )
19 cardid2 7832 . . . . . 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 7241 . . . . . 6  |-  ( (
card `  A )  ~~  A  ->  ( (
card `  A )  ~<_  ( card `  B )  <->  A  ~<_  ( card `  B
) ) )
22 domen2 7242 . . . . . 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 2676 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 1652    e. wcel 1725    =/= wne 2598   _Vcvv 2948    C_ wss 3312   (/)c0 3620   class class class wbr 4204   dom cdm 4870   ` cfv 5446    ~~ cen 7098    ~<_ cdom 7099   cardccrd 7814
This theorem is referenced by:  carddom2  7856
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-er 6897  df-en 7102  df-dom 7103  df-card 7818
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