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Theorem 00lss 17001
Description: The empty structure has no subspaces (for use with fvco4i 5766). (Contributed by Stefan O'Rear, 31-Mar-2015.)
Assertion
Ref Expression
00lss  |-  (/)  =  (
LSubSp `  (/) )

Proof of Theorem 00lss
StepHypRef Expression
1 noel 3638 . . 3  |-  -.  a  e.  (/)
2 base0 14209 . . . . . 6  |-  (/)  =  (
Base `  (/) )
3 eqid 2441 . . . . . 6  |-  ( LSubSp `  (/) )  =  ( LSubSp `
 (/) )
42, 3lssss 16996 . . . . 5  |-  ( a  e.  ( LSubSp `  (/) )  -> 
a  C_  (/) )
5 ss0 3665 . . . . 5  |-  ( a 
C_  (/)  ->  a  =  (/) )
64, 5syl 16 . . . 4  |-  ( a  e.  ( LSubSp `  (/) )  -> 
a  =  (/) )
73lssn0 17000 . . . . 5  |-  ( a  e.  ( LSubSp `  (/) )  -> 
a  =/=  (/) )
87neneqd 2622 . . . 4  |-  ( a  e.  ( LSubSp `  (/) )  ->  -.  a  =  (/) )
96, 8pm2.65i 173 . . 3  |-  -.  a  e.  ( LSubSp `  (/) )
101, 92false 350 . 2  |-  ( a  e.  (/)  <->  a  e.  (
LSubSp `  (/) ) )
1110eqriv 2438 1  |-  (/)  =  (
LSubSp `  (/) )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1364    e. wcel 1761    C_ wss 3325   (/)c0 3634   ` cfv 5415   LSubSpclss 16991
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3184  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-iota 5378  df-fun 5417  df-fv 5423  df-ov 6093  df-slot 14174  df-base 14175  df-lss 16992
This theorem is referenced by:  00lsp  17040  lidlval  17251
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