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Theorem 00lss 17366
Description: The empty structure has no subspaces (for use with fvco4i 5938). (Contributed by Stefan O'Rear, 31-Mar-2015.)
Assertion
Ref Expression
00lss  |-  (/)  =  (
LSubSp `  (/) )

Proof of Theorem 00lss
StepHypRef Expression
1 noel 3784 . . 3  |-  -.  a  e.  (/)
2 base0 14520 . . . . . 6  |-  (/)  =  (
Base `  (/) )
3 eqid 2462 . . . . . 6  |-  ( LSubSp `  (/) )  =  ( LSubSp `
 (/) )
42, 3lssss 17361 . . . . 5  |-  ( a  e.  ( LSubSp `  (/) )  -> 
a  C_  (/) )
5 ss0 3811 . . . . 5  |-  ( a 
C_  (/)  ->  a  =  (/) )
64, 5syl 16 . . . 4  |-  ( a  e.  ( LSubSp `  (/) )  -> 
a  =  (/) )
73lssn0 17365 . . . . 5  |-  ( a  e.  ( LSubSp `  (/) )  -> 
a  =/=  (/) )
87neneqd 2664 . . . 4  |-  ( a  e.  ( LSubSp `  (/) )  ->  -.  a  =  (/) )
96, 8pm2.65i 173 . . 3  |-  -.  a  e.  ( LSubSp `  (/) )
101, 92false 350 . 2  |-  ( a  e.  (/)  <->  a  e.  (
LSubSp `  (/) ) )
1110eqriv 2458 1  |-  (/)  =  (
LSubSp `  (/) )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1374    e. wcel 1762    C_ wss 3471   (/)c0 3780   ` cfv 5581   LSubSpclss 17356
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 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-sbc 3327  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-iota 5544  df-fun 5583  df-fv 5589  df-ov 6280  df-slot 14485  df-base 14486  df-lss 17357
This theorem is referenced by:  00lsp  17405  lidlval  17616
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