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Theorem lmodbn0 15923
Description: The base set of a left module is nonempty. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
Hypothesis
Ref Expression
lmodbn0.b  |-  B  =  ( Base `  W
)
Assertion
Ref Expression
lmodbn0  |-  ( W  e.  LMod  ->  B  =/=  (/) )

Proof of Theorem lmodbn0
StepHypRef Expression
1 lmodgrp 15920 . 2  |-  ( W  e.  LMod  ->  W  e. 
Grp )
2 lmodbn0.b . . 3  |-  B  =  ( Base `  W
)
32grpbn0 14797 . 2  |-  ( W  e.  Grp  ->  B  =/=  (/) )
41, 3syl 16 1  |-  ( W  e.  LMod  ->  B  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1721    =/= wne 2575   (/)c0 3596   ` cfv 5421   Basecbs 13432   Grpcgrp 14648   LModclmod 15913
This theorem is referenced by:  lss1  15978
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-iota 5385  df-fun 5423  df-fv 5429  df-ov 6051  df-riota 6516  df-0g 13690  df-mnd 14653  df-grp 14775  df-lmod 15915
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