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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 16287, and so are equal by mrieqv2d 16299.) (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 16287 | . . . . 5 ⊢ (𝜑 → (𝑁‘𝑆) = (𝑁‘𝑇)) |
6 | pssne 3703 | . . . . . . 7 ⊢ ((𝑁‘𝑇) ⊊ (𝑁‘𝑆) → (𝑁‘𝑇) ≠ (𝑁‘𝑆)) | |
7 | 6 | necomd 2849 | . . . . . 6 ⊢ ((𝑁‘𝑇) ⊊ (𝑁‘𝑆) → (𝑁‘𝑆) ≠ (𝑁‘𝑇)) |
8 | 7 | necon2bi 2824 | . . . . 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 16296 | . . . . . . 7 ⊢ (𝜑 → 𝑆 ⊆ 𝑋) |
13 | 1, 2, 11, 12 | mrieqv2d 16299 | . . . . . 6 ⊢ (𝜑 → (𝑆 ∈ 𝐼 ↔ ∀𝑠(𝑠 ⊊ 𝑆 → (𝑁‘𝑠) ⊊ (𝑁‘𝑆)))) |
14 | 10, 13 | mpbid 222 | . . . . 5 ⊢ (𝜑 → ∀𝑠(𝑠 ⊊ 𝑆 → (𝑁‘𝑠) ⊊ (𝑁‘𝑆))) |
15 | 10, 4 | ssexd 4805 | . . . . . 6 ⊢ (𝜑 → 𝑇 ∈ V) |
16 | simpr 477 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 = 𝑇) → 𝑠 = 𝑇) | |
17 | 16 | psseq1d 3699 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 = 𝑇) → (𝑠 ⊊ 𝑆 ↔ 𝑇 ⊊ 𝑆)) |
18 | 16 | fveq2d 6195 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 = 𝑇) → (𝑁‘𝑠) = (𝑁‘𝑇)) |
19 | 18 | psseq1d 3699 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 = 𝑇) → ((𝑁‘𝑠) ⊊ (𝑁‘𝑆) ↔ (𝑁‘𝑇) ⊊ (𝑁‘𝑆))) |
20 | 17, 19 | imbi12d 334 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑠 = 𝑇) → ((𝑠 ⊊ 𝑆 → (𝑁‘𝑠) ⊊ (𝑁‘𝑆)) ↔ (𝑇 ⊊ 𝑆 → (𝑁‘𝑇) ⊊ (𝑁‘𝑆)))) |
21 | 15, 20 | spcdv 3291 | . . . . 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 189 | . . 3 ⊢ (𝜑 → ¬ 𝑇 ⊊ 𝑆) |
24 | sspss 3706 | . . . . 5 ⊢ (𝑇 ⊆ 𝑆 ↔ (𝑇 ⊊ 𝑆 ∨ 𝑇 = 𝑆)) | |
25 | 4, 24 | sylib 208 | . . . 4 ⊢ (𝜑 → (𝑇 ⊊ 𝑆 ∨ 𝑇 = 𝑆)) |
26 | 25 | ord 392 | . . 3 ⊢ (𝜑 → (¬ 𝑇 ⊊ 𝑆 → 𝑇 = 𝑆)) |
27 | 23, 26 | mpd 15 | . 2 ⊢ (𝜑 → 𝑇 = 𝑆) |
28 | 27 | eqcomd 2628 | 1 ⊢ (𝜑 → 𝑆 = 𝑇) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 383 ∧ wa 384 ∀wal 1481 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ⊆ wss 3574 ⊊ wpss 3575 ‘cfv 5888 Moorecmre 16242 mrClscmrc 16243 mrIndcmri 16244 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-int 4476 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fv 5896 df-mre 16246 df-mrc 16247 df-mri 16248 |
This theorem is referenced by: mreexexlem3d 16306 acsmap2d 17179 |
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