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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 17670, and so are equal by mrieqv2d 17682.) (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 17670 | . . . . 5 ⊢ (𝜑 → (𝑁‘𝑆) = (𝑁‘𝑇)) | 
| 6 | pssne 4099 | . . . . . . 7 ⊢ ((𝑁‘𝑇) ⊊ (𝑁‘𝑆) → (𝑁‘𝑇) ≠ (𝑁‘𝑆)) | |
| 7 | 6 | necomd 2996 | . . . . . 6 ⊢ ((𝑁‘𝑇) ⊊ (𝑁‘𝑆) → (𝑁‘𝑆) ≠ (𝑁‘𝑇)) | 
| 8 | 7 | necon2bi 2971 | . . . . 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 17679 | . . . . . . 7 ⊢ (𝜑 → 𝑆 ⊆ 𝑋) | 
| 13 | 1, 2, 11, 12 | mrieqv2d 17682 | . . . . . 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 5324 | . . . . . 6 ⊢ (𝜑 → 𝑇 ∈ V) | 
| 16 | simpr 484 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 = 𝑇) → 𝑠 = 𝑇) | |
| 17 | 16 | psseq1d 4095 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 = 𝑇) → (𝑠 ⊊ 𝑆 ↔ 𝑇 ⊊ 𝑆)) | 
| 18 | 16 | fveq2d 6910 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 = 𝑇) → (𝑁‘𝑠) = (𝑁‘𝑇)) | 
| 19 | 18 | psseq1d 4095 | . . . . . . 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 4102 | . . . . 5 ⊢ (𝑇 ⊆ 𝑆 ↔ (𝑇 ⊊ 𝑆 ∨ 𝑇 = 𝑆)) | |
| 25 | 4, 24 | sylib 218 | . . . 4 ⊢ (𝜑 → (𝑇 ⊊ 𝑆 ∨ 𝑇 = 𝑆)) | 
| 26 | 25 | ord 865 | . . 3 ⊢ (𝜑 → (¬ 𝑇 ⊊ 𝑆 → 𝑇 = 𝑆)) | 
| 27 | 23, 26 | mpd 15 | . 2 ⊢ (𝜑 → 𝑇 = 𝑆) | 
| 28 | 27 | eqcomd 2743 | 1 ⊢ (𝜑 → 𝑆 = 𝑇) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∨ wo 848 ∀wal 1538 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ⊆ wss 3951 ⊊ wpss 3952 ‘cfv 6561 Moorecmre 17625 mrClscmrc 17626 mrIndcmri 17627 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-mre 17629 df-mrc 17630 df-mri 17631 | 
| This theorem is referenced by: mreexexlem3d 17689 acsmap2d 18600 | 
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