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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 6186). (Contributed by Stefan O'Rear, 31-Mar-2015.) |
Ref | Expression |
---|---|
00lss | ⊢ ∅ = (LSubSp‘∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | noel 3878 | . . 3 ⊢ ¬ 𝑎 ∈ ∅ | |
2 | base0 15740 | . . . . . 6 ⊢ ∅ = (Base‘∅) | |
3 | eqid 2610 | . . . . . 6 ⊢ (LSubSp‘∅) = (LSubSp‘∅) | |
4 | 2, 3 | lssss 18758 | . . . . 5 ⊢ (𝑎 ∈ (LSubSp‘∅) → 𝑎 ⊆ ∅) |
5 | ss0 3926 | . . . . 5 ⊢ (𝑎 ⊆ ∅ → 𝑎 = ∅) | |
6 | 4, 5 | syl 17 | . . . 4 ⊢ (𝑎 ∈ (LSubSp‘∅) → 𝑎 = ∅) |
7 | 3 | lssn0 18762 | . . . . 5 ⊢ (𝑎 ∈ (LSubSp‘∅) → 𝑎 ≠ ∅) |
8 | 7 | neneqd 2787 | . . . 4 ⊢ (𝑎 ∈ (LSubSp‘∅) → ¬ 𝑎 = ∅) |
9 | 6, 8 | pm2.65i 184 | . . 3 ⊢ ¬ 𝑎 ∈ (LSubSp‘∅) |
10 | 1, 9 | 2false 364 | . 2 ⊢ (𝑎 ∈ ∅ ↔ 𝑎 ∈ (LSubSp‘∅)) |
11 | 10 | eqriv 2607 | 1 ⊢ ∅ = (LSubSp‘∅) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∈ wcel 1977 ⊆ wss 3540 ∅c0 3874 ‘cfv 5804 LSubSpclss 18753 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-ov 6552 df-slot 15699 df-base 15700 df-lss 18754 |
This theorem is referenced by: 00lsp 18802 lidlval 19013 |
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