Theorem mrissmrcd 16905

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 16892, and so are equal by mrieqv2d 16904.) (Contributed by David Moews, 1-May-2017.)

Hypotheses

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)

Assertion

Ref	Expression
mrissmrcd	⊢ (𝜑 → 𝑆 = 𝑇)

Proof of Theorem mrissmrcd

Dummy variable 𝑠 is distinct from all other variables.

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 16892	. . . . 5 ⊢ (𝜑 → (𝑁‘𝑆) = (𝑁‘𝑇))
6		pssne 4072	. . . . . . 7 ⊢ ((𝑁‘𝑇) ⊊ (𝑁‘𝑆) → (𝑁‘𝑇) ≠ (𝑁‘𝑆))
7	6	necomd 3071	. . . . . 6 ⊢ ((𝑁‘𝑇) ⊊ (𝑁‘𝑆) → (𝑁‘𝑆) ≠ (𝑁‘𝑇))
8	7	necon2bi 3046	. . . . 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 16901	. . . . . . 7 ⊢ (𝜑 → 𝑆 ⊆ 𝑋)
13	1, 2, 11, 12	mrieqv2d 16904	. . . . . 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 5220	. . . . . 6 ⊢ (𝜑 → 𝑇 ∈ V)
16		simpr 487	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 = 𝑇) → 𝑠 = 𝑇)
17	16	psseq1d 4068	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 = 𝑇) → (𝑠 ⊊ 𝑆 ↔ 𝑇 ⊊ 𝑆))
18	16	fveq2d 6668	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 = 𝑇) → (𝑁‘𝑠) = (𝑁‘𝑇))
19	18	psseq1d 4068	. . . . . . 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 3592	. . . . 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 4075	. . . . 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 2827	1 ⊢ (𝜑 → 𝑆 = 𝑇)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   ∨ wo 843  ∀wal 1531   = wceq 1533   ∈ wcel 2110  Vcvv 3494   ⊆ wss 3935   ⊊ wpss 3936  ‘cfv 6349  Moorecmre 16847  mrClscmrc 16848  mrIndcmri 16849

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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-int 4869  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-fv 6357  df-mre 16851  df-mrc 16852  df-mri 16853

This theorem is referenced by:  mreexexlem3d  16911  acsmap2d  17783