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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 6276). (Contributed by Stefan O'Rear, 31-Mar-2015.) |
Ref | Expression |
---|---|
00lss | ⊢ ∅ = (LSubSp‘∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | noel 3919 | . . 3 ⊢ ¬ 𝑎 ∈ ∅ | |
2 | base0 15912 | . . . . . 6 ⊢ ∅ = (Base‘∅) | |
3 | eqid 2622 | . . . . . 6 ⊢ (LSubSp‘∅) = (LSubSp‘∅) | |
4 | 2, 3 | lssss 18937 | . . . . 5 ⊢ (𝑎 ∈ (LSubSp‘∅) → 𝑎 ⊆ ∅) |
5 | ss0 3974 | . . . . 5 ⊢ (𝑎 ⊆ ∅ → 𝑎 = ∅) | |
6 | 4, 5 | syl 17 | . . . 4 ⊢ (𝑎 ∈ (LSubSp‘∅) → 𝑎 = ∅) |
7 | 3 | lssn0 18941 | . . . . 5 ⊢ (𝑎 ∈ (LSubSp‘∅) → 𝑎 ≠ ∅) |
8 | 7 | neneqd 2799 | . . . 4 ⊢ (𝑎 ∈ (LSubSp‘∅) → ¬ 𝑎 = ∅) |
9 | 6, 8 | pm2.65i 185 | . . 3 ⊢ ¬ 𝑎 ∈ (LSubSp‘∅) |
10 | 1, 9 | 2false 365 | . 2 ⊢ (𝑎 ∈ ∅ ↔ 𝑎 ∈ (LSubSp‘∅)) |
11 | 10 | eqriv 2619 | 1 ⊢ ∅ = (LSubSp‘∅) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1483 ∈ wcel 1990 ⊆ wss 3574 ∅c0 3915 ‘cfv 5888 LSubSpclss 18932 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 df-ov 6653 df-slot 15861 df-base 15863 df-lss 18933 |
This theorem is referenced by: 00lsp 18981 lidlval 19192 |
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