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Theorem 00lss 19305
Description: The empty structure has no subspaces (for use with fvco4i 6527). (Contributed by Stefan O'Rear, 31-Mar-2015.)
Assertion
Ref Expression
00lss ∅ = (LSubSp‘∅)

Proof of Theorem 00lss
StepHypRef Expression
1 noel 4150 . . 3 ¬ 𝑎 ∈ ∅
2 base0 16282 . . . . . 6 ∅ = (Base‘∅)
3 eqid 2825 . . . . . 6 (LSubSp‘∅) = (LSubSp‘∅)
42, 3lssss 19300 . . . . 5 (𝑎 ∈ (LSubSp‘∅) → 𝑎 ⊆ ∅)
5 ss0 4201 . . . . 5 (𝑎 ⊆ ∅ → 𝑎 = ∅)
64, 5syl 17 . . . 4 (𝑎 ∈ (LSubSp‘∅) → 𝑎 = ∅)
73lssn0 19304 . . . . 5 (𝑎 ∈ (LSubSp‘∅) → 𝑎 ≠ ∅)
87neneqd 3004 . . . 4 (𝑎 ∈ (LSubSp‘∅) → ¬ 𝑎 = ∅)
96, 8pm2.65i 186 . . 3 ¬ 𝑎 ∈ (LSubSp‘∅)
101, 92false 367 . 2 (𝑎 ∈ ∅ ↔ 𝑎 ∈ (LSubSp‘∅))
1110eqriv 2822 1 ∅ = (LSubSp‘∅)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1656  wcel 2164  wss 3798  c0 4146  cfv 6127  LSubSpclss 19295
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-iota 6090  df-fun 6129  df-fv 6135  df-ov 6913  df-slot 16233  df-base 16235  df-lss 19296
This theorem is referenced by:  00lsp  19347  lidlval  19560
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