![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > mrieqvd | Structured version Visualization version GIF version |
Description: In a Moore system, a set is independent if and only if, for all elements of the set, the closure of the set with the element removed is unequal to the closure of the original set. Part of Proposition 4.1.3 in [FaureFrolicher] p. 83. (Contributed by David Moews, 1-May-2017.) |
Ref | Expression |
---|---|
mrieqvd.1 | ⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋)) |
mrieqvd.2 | ⊢ 𝑁 = (mrCls‘𝐴) |
mrieqvd.3 | ⊢ 𝐼 = (mrInd‘𝐴) |
mrieqvd.4 | ⊢ (𝜑 → 𝑆 ⊆ 𝑋) |
Ref | Expression |
---|---|
mrieqvd | ⊢ (𝜑 → (𝑆 ∈ 𝐼 ↔ ∀𝑥 ∈ 𝑆 (𝑁‘(𝑆 ∖ {𝑥})) ≠ (𝑁‘𝑆))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mrieqvd.2 | . . 3 ⊢ 𝑁 = (mrCls‘𝐴) | |
2 | mrieqvd.3 | . . 3 ⊢ 𝐼 = (mrInd‘𝐴) | |
3 | mrieqvd.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋)) | |
4 | mrieqvd.4 | . . 3 ⊢ (𝜑 → 𝑆 ⊆ 𝑋) | |
5 | 1, 2, 3, 4 | ismri2d 16653 | . 2 ⊢ (𝜑 → (𝑆 ∈ 𝐼 ↔ ∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})))) |
6 | 3 | adantr 474 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐴 ∈ (Moore‘𝑋)) |
7 | 4 | adantr 474 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑆 ⊆ 𝑋) |
8 | simpr 479 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝑆) | |
9 | 6, 1, 7, 8 | mrieqvlemd 16649 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})) ↔ (𝑁‘(𝑆 ∖ {𝑥})) = (𝑁‘𝑆))) |
10 | 9 | necon3bbid 3036 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})) ↔ (𝑁‘(𝑆 ∖ {𝑥})) ≠ (𝑁‘𝑆))) |
11 | 10 | ralbidva 3194 | . 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})) ↔ ∀𝑥 ∈ 𝑆 (𝑁‘(𝑆 ∖ {𝑥})) ≠ (𝑁‘𝑆))) |
12 | 5, 11 | bitrd 271 | 1 ⊢ (𝜑 → (𝑆 ∈ 𝐼 ↔ ∀𝑥 ∈ 𝑆 (𝑁‘(𝑆 ∖ {𝑥})) ≠ (𝑁‘𝑆))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1656 ∈ wcel 2164 ≠ wne 2999 ∀wral 3117 ∖ cdif 3795 ⊆ wss 3798 {csn 4399 ‘cfv 6127 Moorecmre 16602 mrClscmrc 16603 mrIndcmri 16604 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-int 4700 df-br 4876 df-opab 4938 df-mpt 4955 df-id 5252 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-fv 6135 df-mre 16606 df-mrc 16607 df-mri 16608 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |