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Mirrors > Home > MPE Home > Th. List > mrissmrcd | Structured version Visualization version GIF version |
Description: In a Moore system, if an independent set is between a set and its closure, the two sets are equal (since the two sets must have equal closures by mressmrcd 17672, and so are equal by mrieqv2d 17684.) (Contributed by David Moews, 1-May-2017.) |
Ref | Expression |
---|---|
mrissmrcd.1 | ⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋)) |
mrissmrcd.2 | ⊢ 𝑁 = (mrCls‘𝐴) |
mrissmrcd.3 | ⊢ 𝐼 = (mrInd‘𝐴) |
mrissmrcd.4 | ⊢ (𝜑 → 𝑆 ⊆ (𝑁‘𝑇)) |
mrissmrcd.5 | ⊢ (𝜑 → 𝑇 ⊆ 𝑆) |
mrissmrcd.6 | ⊢ (𝜑 → 𝑆 ∈ 𝐼) |
Ref | Expression |
---|---|
mrissmrcd | ⊢ (𝜑 → 𝑆 = 𝑇) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mrissmrcd.1 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋)) | |
2 | mrissmrcd.2 | . . . . . 6 ⊢ 𝑁 = (mrCls‘𝐴) | |
3 | mrissmrcd.4 | . . . . . 6 ⊢ (𝜑 → 𝑆 ⊆ (𝑁‘𝑇)) | |
4 | mrissmrcd.5 | . . . . . 6 ⊢ (𝜑 → 𝑇 ⊆ 𝑆) | |
5 | 1, 2, 3, 4 | mressmrcd 17672 | . . . . 5 ⊢ (𝜑 → (𝑁‘𝑆) = (𝑁‘𝑇)) |
6 | pssne 4109 | . . . . . . 7 ⊢ ((𝑁‘𝑇) ⊊ (𝑁‘𝑆) → (𝑁‘𝑇) ≠ (𝑁‘𝑆)) | |
7 | 6 | necomd 2994 | . . . . . 6 ⊢ ((𝑁‘𝑇) ⊊ (𝑁‘𝑆) → (𝑁‘𝑆) ≠ (𝑁‘𝑇)) |
8 | 7 | necon2bi 2969 | . . . . 5 ⊢ ((𝑁‘𝑆) = (𝑁‘𝑇) → ¬ (𝑁‘𝑇) ⊊ (𝑁‘𝑆)) |
9 | 5, 8 | syl 17 | . . . 4 ⊢ (𝜑 → ¬ (𝑁‘𝑇) ⊊ (𝑁‘𝑆)) |
10 | mrissmrcd.6 | . . . . . 6 ⊢ (𝜑 → 𝑆 ∈ 𝐼) | |
11 | mrissmrcd.3 | . . . . . . 7 ⊢ 𝐼 = (mrInd‘𝐴) | |
12 | 11, 1, 10 | mrissd 17681 | . . . . . . 7 ⊢ (𝜑 → 𝑆 ⊆ 𝑋) |
13 | 1, 2, 11, 12 | mrieqv2d 17684 | . . . . . 6 ⊢ (𝜑 → (𝑆 ∈ 𝐼 ↔ ∀𝑠(𝑠 ⊊ 𝑆 → (𝑁‘𝑠) ⊊ (𝑁‘𝑆)))) |
14 | 10, 13 | mpbid 232 | . . . . 5 ⊢ (𝜑 → ∀𝑠(𝑠 ⊊ 𝑆 → (𝑁‘𝑠) ⊊ (𝑁‘𝑆))) |
15 | 10, 4 | ssexd 5330 | . . . . . 6 ⊢ (𝜑 → 𝑇 ∈ V) |
16 | simpr 484 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 = 𝑇) → 𝑠 = 𝑇) | |
17 | 16 | psseq1d 4105 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 = 𝑇) → (𝑠 ⊊ 𝑆 ↔ 𝑇 ⊊ 𝑆)) |
18 | 16 | fveq2d 6911 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 = 𝑇) → (𝑁‘𝑠) = (𝑁‘𝑇)) |
19 | 18 | psseq1d 4105 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 = 𝑇) → ((𝑁‘𝑠) ⊊ (𝑁‘𝑆) ↔ (𝑁‘𝑇) ⊊ (𝑁‘𝑆))) |
20 | 17, 19 | imbi12d 344 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑠 = 𝑇) → ((𝑠 ⊊ 𝑆 → (𝑁‘𝑠) ⊊ (𝑁‘𝑆)) ↔ (𝑇 ⊊ 𝑆 → (𝑁‘𝑇) ⊊ (𝑁‘𝑆)))) |
21 | 15, 20 | spcdv 3594 | . . . . 5 ⊢ (𝜑 → (∀𝑠(𝑠 ⊊ 𝑆 → (𝑁‘𝑠) ⊊ (𝑁‘𝑆)) → (𝑇 ⊊ 𝑆 → (𝑁‘𝑇) ⊊ (𝑁‘𝑆)))) |
22 | 14, 21 | mpd 15 | . . . 4 ⊢ (𝜑 → (𝑇 ⊊ 𝑆 → (𝑁‘𝑇) ⊊ (𝑁‘𝑆))) |
23 | 9, 22 | mtod 198 | . . 3 ⊢ (𝜑 → ¬ 𝑇 ⊊ 𝑆) |
24 | sspss 4112 | . . . . 5 ⊢ (𝑇 ⊆ 𝑆 ↔ (𝑇 ⊊ 𝑆 ∨ 𝑇 = 𝑆)) | |
25 | 4, 24 | sylib 218 | . . . 4 ⊢ (𝜑 → (𝑇 ⊊ 𝑆 ∨ 𝑇 = 𝑆)) |
26 | 25 | ord 864 | . . 3 ⊢ (𝜑 → (¬ 𝑇 ⊊ 𝑆 → 𝑇 = 𝑆)) |
27 | 23, 26 | mpd 15 | . 2 ⊢ (𝜑 → 𝑇 = 𝑆) |
28 | 27 | eqcomd 2741 | 1 ⊢ (𝜑 → 𝑆 = 𝑇) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∨ wo 847 ∀wal 1535 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ⊆ wss 3963 ⊊ wpss 3964 ‘cfv 6563 Moorecmre 17627 mrClscmrc 17628 mrIndcmri 17629 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fv 6571 df-mre 17631 df-mrc 17632 df-mri 17633 |
This theorem is referenced by: mreexexlem3d 17691 acsmap2d 18613 |
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