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Theorem sdomsdomcardi 8369
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 7668 . . . . 5  |-  -.  A  ~< 
(/)
2 ndmfv 5896 . . . . . 6  |-  ( -.  B  e.  dom  card  -> 
( card `  B )  =  (/) )
32breq2d 4468 . . . . 5  |-  ( -.  B  e.  dom  card  -> 
( A  ~<  ( card `  B )  <->  A  ~<  (/) ) )
41, 3mtbiri 303 . . . 4  |-  ( -.  B  e.  dom  card  ->  -.  A  ~<  ( card `  B ) )
54con4i 130 . . 3  |-  ( A 
~<  ( card `  B
)  ->  B  e.  dom  card )
6 cardid2 8351 . . 3  |-  ( B  e.  dom  card  ->  (
card `  B )  ~~  B )
75, 6syl 16 . 2  |-  ( A 
~<  ( card `  B
)  ->  ( card `  B )  ~~  B
)
8 sdomentr 7670 . 2  |-  ( ( A  ~<  ( card `  B )  /\  ( card `  B )  ~~  B )  ->  A  ~<  B )
97, 8mpdan 668 1  |-  ( A 
~<  ( card `  B
)  ->  A  ~<  B )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    e. wcel 1819   (/)c0 3793   class class class wbr 4456   dom cdm 5008   ` cfv 5594    ~~ cen 7532    ~< csdm 7534   cardccrd 8333
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-card 8337
This theorem is referenced by:  sdomsdomcard  8952
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