![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > 00lss | Structured version Visualization version GIF version |
Description: The empty structure has no subspaces (for use with fvco4i 6995). (Contributed by Stefan O'Rear, 31-Mar-2015.) |
Ref | Expression |
---|---|
00lss | ⊢ ∅ = (LSubSp‘∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | noel 4330 | . . 3 ⊢ ¬ 𝑎 ∈ ∅ | |
2 | base0 17213 | . . . . . 6 ⊢ ∅ = (Base‘∅) | |
3 | eqid 2726 | . . . . . 6 ⊢ (LSubSp‘∅) = (LSubSp‘∅) | |
4 | 2, 3 | lssss 20909 | . . . . 5 ⊢ (𝑎 ∈ (LSubSp‘∅) → 𝑎 ⊆ ∅) |
5 | ss0 4396 | . . . . 5 ⊢ (𝑎 ⊆ ∅ → 𝑎 = ∅) | |
6 | 4, 5 | syl 17 | . . . 4 ⊢ (𝑎 ∈ (LSubSp‘∅) → 𝑎 = ∅) |
7 | 3 | lssn0 20913 | . . . . 5 ⊢ (𝑎 ∈ (LSubSp‘∅) → 𝑎 ≠ ∅) |
8 | 7 | neneqd 2935 | . . . 4 ⊢ (𝑎 ∈ (LSubSp‘∅) → ¬ 𝑎 = ∅) |
9 | 6, 8 | pm2.65i 193 | . . 3 ⊢ ¬ 𝑎 ∈ (LSubSp‘∅) |
10 | 1, 9 | 2false 374 | . 2 ⊢ (𝑎 ∈ ∅ ↔ 𝑎 ∈ (LSubSp‘∅)) |
11 | 10 | eqriv 2723 | 1 ⊢ ∅ = (LSubSp‘∅) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 ∈ wcel 2099 ⊆ wss 3946 ∅c0 4322 ‘cfv 6546 LSubSpclss 20904 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 ax-cnex 11205 ax-1cn 11207 ax-addcl 11209 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-iun 4995 df-br 5146 df-opab 5208 df-mpt 5229 df-tr 5263 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6304 df-ord 6371 df-on 6372 df-lim 6373 df-suc 6374 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-ov 7419 df-om 7869 df-2nd 7996 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-rdg 8432 df-nn 12259 df-slot 17179 df-ndx 17191 df-base 17209 df-lss 20905 |
This theorem is referenced by: 00lsp 20954 lidlval 21195 |
Copyright terms: Public domain | W3C validator |