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Theorem sdomsdomcardi 8423
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 7722 . . . . 5  |-  -.  A  ~< 
(/)
2 ndmfv 5903 . . . . . 6  |-  ( -.  B  e.  dom  card  -> 
( card `  B )  =  (/) )
32breq2d 4407 . . . . 5  |-  ( -.  B  e.  dom  card  -> 
( A  ~<  ( card `  B )  <->  A  ~<  (/) ) )
41, 3mtbiri 310 . . . 4  |-  ( -.  B  e.  dom  card  ->  -.  A  ~<  ( card `  B ) )
54con4i 135 . . 3  |-  ( A 
~<  ( card `  B
)  ->  B  e.  dom  card )
6 cardid2 8405 . . 3  |-  ( B  e.  dom  card  ->  (
card `  B )  ~~  B )
75, 6syl 17 . 2  |-  ( A 
~<  ( card `  B
)  ->  ( card `  B )  ~~  B
)
8 sdomentr 7724 . 2  |-  ( ( A  ~<  ( card `  B )  /\  ( card `  B )  ~~  B )  ->  A  ~<  B )
97, 8mpdan 681 1  |-  ( A 
~<  ( card `  B
)  ->  A  ~<  B )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    e. wcel 1904   (/)c0 3722   class class class wbr 4395   dom cdm 4839   ` cfv 5589    ~~ cen 7584    ~< csdm 7586   cardccrd 8387
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-ord 5433  df-on 5434  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-card 8391
This theorem is referenced by:  sdomsdomcard  9003
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