MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sdomsdomcardi Unicode version

Theorem sdomsdomcardi 7784
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 7168 . . . . 5  |-  -.  A  ~< 
(/)
2 ndmfv 5688 . . . . . 6  |-  ( -.  B  e.  dom  card  -> 
( card `  B )  =  (/) )
32breq2d 4158 . . . . 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 7766 . . 3  |-  ( B  e.  dom  card  ->  (
card `  B )  ~~  B )
75, 6syl 16 . 2  |-  ( A 
~<  ( card `  B
)  ->  ( card `  B )  ~~  B
)
8 sdomentr 7170 . 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 1717   (/)c0 3564   class class class wbr 4146   dom cdm 4811   ` cfv 5387    ~~ cen 7035    ~< csdm 7037   cardccrd 7748
This theorem is referenced by:  sdomsdomcard  8361
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 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
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 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-sbc 3098  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-int 3986  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-er 6834  df-en 7039  df-dom 7040  df-sdom 7041  df-card 7752
  Copyright terms: Public domain W3C validator