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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 16900, and so are equal by mrieqv2d 16912.) (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 16900 | . . . . 5 ⊢ (𝜑 → (𝑁‘𝑆) = (𝑁‘𝑇)) |
6 | pssne 4075 | . . . . . . 7 ⊢ ((𝑁‘𝑇) ⊊ (𝑁‘𝑆) → (𝑁‘𝑇) ≠ (𝑁‘𝑆)) | |
7 | 6 | necomd 3073 | . . . . . 6 ⊢ ((𝑁‘𝑇) ⊊ (𝑁‘𝑆) → (𝑁‘𝑆) ≠ (𝑁‘𝑇)) |
8 | 7 | necon2bi 3048 | . . . . 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 16909 | . . . . . . 7 ⊢ (𝜑 → 𝑆 ⊆ 𝑋) |
13 | 1, 2, 11, 12 | mrieqv2d 16912 | . . . . . 6 ⊢ (𝜑 → (𝑆 ∈ 𝐼 ↔ ∀𝑠(𝑠 ⊊ 𝑆 → (𝑁‘𝑠) ⊊ (𝑁‘𝑆)))) |
14 | 10, 13 | mpbid 234 | . . . . 5 ⊢ (𝜑 → ∀𝑠(𝑠 ⊊ 𝑆 → (𝑁‘𝑠) ⊊ (𝑁‘𝑆))) |
15 | 10, 4 | ssexd 5230 | . . . . . 6 ⊢ (𝜑 → 𝑇 ∈ V) |
16 | simpr 487 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 = 𝑇) → 𝑠 = 𝑇) | |
17 | 16 | psseq1d 4071 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 = 𝑇) → (𝑠 ⊊ 𝑆 ↔ 𝑇 ⊊ 𝑆)) |
18 | 16 | fveq2d 6676 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 = 𝑇) → (𝑁‘𝑠) = (𝑁‘𝑇)) |
19 | 18 | psseq1d 4071 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 = 𝑇) → ((𝑁‘𝑠) ⊊ (𝑁‘𝑆) ↔ (𝑁‘𝑇) ⊊ (𝑁‘𝑆))) |
20 | 17, 19 | imbi12d 347 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑠 = 𝑇) → ((𝑠 ⊊ 𝑆 → (𝑁‘𝑠) ⊊ (𝑁‘𝑆)) ↔ (𝑇 ⊊ 𝑆 → (𝑁‘𝑇) ⊊ (𝑁‘𝑆)))) |
21 | 15, 20 | spcdv 3595 | . . . . 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 200 | . . 3 ⊢ (𝜑 → ¬ 𝑇 ⊊ 𝑆) |
24 | sspss 4078 | . . . . 5 ⊢ (𝑇 ⊆ 𝑆 ↔ (𝑇 ⊊ 𝑆 ∨ 𝑇 = 𝑆)) | |
25 | 4, 24 | sylib 220 | . . . 4 ⊢ (𝜑 → (𝑇 ⊊ 𝑆 ∨ 𝑇 = 𝑆)) |
26 | 25 | ord 860 | . . 3 ⊢ (𝜑 → (¬ 𝑇 ⊊ 𝑆 → 𝑇 = 𝑆)) |
27 | 23, 26 | mpd 15 | . 2 ⊢ (𝜑 → 𝑇 = 𝑆) |
28 | 27 | eqcomd 2829 | 1 ⊢ (𝜑 → 𝑆 = 𝑇) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 ∨ wo 843 ∀wal 1535 = wceq 1537 ∈ wcel 2114 Vcvv 3496 ⊆ wss 3938 ⊊ wpss 3939 ‘cfv 6357 Moorecmre 16855 mrClscmrc 16856 mrIndcmri 16857 |
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 ax-un 7463 |
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-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-int 4879 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-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-fv 6365 df-mre 16859 df-mrc 16860 df-mri 16861 |
This theorem is referenced by: mreexexlem3d 16919 acsmap2d 17791 |
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