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Mirrors > Home > MPE Home > Th. List > 00lss | Structured version Visualization version GIF version |
Description: The empty structure has no subspaces (for use with fvco4i 6739). (Contributed by Stefan O'Rear, 31-Mar-2015.) |
Ref | Expression |
---|---|
00lss | ⊢ ∅ = (LSubSp‘∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | noel 4247 | . . 3 ⊢ ¬ 𝑎 ∈ ∅ | |
2 | base0 16528 | . . . . . 6 ⊢ ∅ = (Base‘∅) | |
3 | eqid 2798 | . . . . . 6 ⊢ (LSubSp‘∅) = (LSubSp‘∅) | |
4 | 2, 3 | lssss 19701 | . . . . 5 ⊢ (𝑎 ∈ (LSubSp‘∅) → 𝑎 ⊆ ∅) |
5 | ss0 4306 | . . . . 5 ⊢ (𝑎 ⊆ ∅ → 𝑎 = ∅) | |
6 | 4, 5 | syl 17 | . . . 4 ⊢ (𝑎 ∈ (LSubSp‘∅) → 𝑎 = ∅) |
7 | 3 | lssn0 19705 | . . . . 5 ⊢ (𝑎 ∈ (LSubSp‘∅) → 𝑎 ≠ ∅) |
8 | 7 | neneqd 2992 | . . . 4 ⊢ (𝑎 ∈ (LSubSp‘∅) → ¬ 𝑎 = ∅) |
9 | 6, 8 | pm2.65i 197 | . . 3 ⊢ ¬ 𝑎 ∈ (LSubSp‘∅) |
10 | 1, 9 | 2false 379 | . 2 ⊢ (𝑎 ∈ ∅ ↔ 𝑎 ∈ (LSubSp‘∅)) |
11 | 10 | eqriv 2795 | 1 ⊢ ∅ = (LSubSp‘∅) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 ∈ wcel 2111 ⊆ wss 3881 ∅c0 4243 ‘cfv 6324 LSubSpclss 19696 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-iota 6283 df-fun 6326 df-fv 6332 df-ov 7138 df-slot 16479 df-base 16481 df-lss 19697 |
This theorem is referenced by: 00lsp 19746 lidlval 19957 |
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