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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 6764). (Contributed by Stefan O'Rear, 31-Mar-2015.) |
Ref | Expression |
---|---|
00lss | ⊢ ∅ = (LSubSp‘∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | noel 4298 | . . 3 ⊢ ¬ 𝑎 ∈ ∅ | |
2 | base0 16538 | . . . . . 6 ⊢ ∅ = (Base‘∅) | |
3 | eqid 2823 | . . . . . 6 ⊢ (LSubSp‘∅) = (LSubSp‘∅) | |
4 | 2, 3 | lssss 19710 | . . . . 5 ⊢ (𝑎 ∈ (LSubSp‘∅) → 𝑎 ⊆ ∅) |
5 | ss0 4354 | . . . . 5 ⊢ (𝑎 ⊆ ∅ → 𝑎 = ∅) | |
6 | 4, 5 | syl 17 | . . . 4 ⊢ (𝑎 ∈ (LSubSp‘∅) → 𝑎 = ∅) |
7 | 3 | lssn0 19714 | . . . . 5 ⊢ (𝑎 ∈ (LSubSp‘∅) → 𝑎 ≠ ∅) |
8 | 7 | neneqd 3023 | . . . 4 ⊢ (𝑎 ∈ (LSubSp‘∅) → ¬ 𝑎 = ∅) |
9 | 6, 8 | pm2.65i 196 | . . 3 ⊢ ¬ 𝑎 ∈ (LSubSp‘∅) |
10 | 1, 9 | 2false 378 | . 2 ⊢ (𝑎 ∈ ∅ ↔ 𝑎 ∈ (LSubSp‘∅)) |
11 | 10 | eqriv 2820 | 1 ⊢ ∅ = (LSubSp‘∅) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2114 ⊆ wss 3938 ∅c0 4293 ‘cfv 6357 LSubSpclss 19705 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-iota 6316 df-fun 6359 df-fv 6365 df-ov 7161 df-slot 16489 df-base 16491 df-lss 19706 |
This theorem is referenced by: 00lsp 19755 lidlval 19966 |
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