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Theorem sdomsdomcardi 7851
Description: A set strictly dominates if its cardinal strictly dominates. (Contributed by Mario Carneiro, 13-Jan-2013.)
Assertion
Ref Expression
sdomsdomcardi  |-  ( A 
~<  ( card `  B
)  ->  A  ~<  B )

Proof of Theorem sdomsdomcardi
StepHypRef Expression
1 sdom0 7232 . . . . 5  |-  -.  A  ~< 
(/)
2 ndmfv 5748 . . . . . 6  |-  ( -.  B  e.  dom  card  -> 
( card `  B )  =  (/) )
32breq2d 4217 . . . . 5  |-  ( -.  B  e.  dom  card  -> 
( A  ~<  ( card `  B )  <->  A  ~<  (/) ) )
41, 3mtbiri 295 . . . 4  |-  ( -.  B  e.  dom  card  ->  -.  A  ~<  ( card `  B ) )
54con4i 124 . . 3  |-  ( A 
~<  ( card `  B
)  ->  B  e.  dom  card )
6 cardid2 7833 . . 3  |-  ( B  e.  dom  card  ->  (
card `  B )  ~~  B )
75, 6syl 16 . 2  |-  ( A 
~<  ( card `  B
)  ->  ( card `  B )  ~~  B
)
8 sdomentr 7234 . 2  |-  ( ( A  ~<  ( card `  B )  /\  ( card `  B )  ~~  B )  ->  A  ~<  B )
97, 8mpdan 650 1  |-  ( A 
~<  ( card `  B
)  ->  A  ~<  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    e. wcel 1725   (/)c0 3621   class class class wbr 4205   dom cdm 4871   ` cfv 5447    ~~ cen 7099    ~< csdm 7101   cardccrd 7815
This theorem is referenced by:  sdomsdomcard  8428
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 2417  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396  ax-un 4694
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2703  df-rex 2704  df-rab 2707  df-v 2951  df-sbc 3155  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-pss 3329  df-nul 3622  df-if 3733  df-pw 3794  df-sn 3813  df-pr 3814  df-tp 3815  df-op 3816  df-uni 4009  df-int 4044  df-br 4206  df-opab 4260  df-mpt 4261  df-tr 4296  df-eprel 4487  df-id 4491  df-po 4496  df-so 4497  df-fr 4534  df-we 4536  df-ord 4577  df-on 4578  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-res 4883  df-ima 4884  df-iota 5411  df-fun 5449  df-fn 5450  df-f 5451  df-f1 5452  df-fo 5453  df-f1o 5454  df-fv 5455  df-er 6898  df-en 7103  df-dom 7104  df-sdom 7105  df-card 7819
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