Theorem mrissmrcd 17673

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 17660, and so are equal by mrieqv2d 17672.) (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 17660	. . . . 5 ⊢ (𝜑 → (𝑁‘𝑆) = (𝑁‘𝑇))
6		pssne 4053	. . . . . . 7 ⊢ ((𝑁‘𝑇) ⊊ (𝑁‘𝑆) → (𝑁‘𝑇) ≠ (𝑁‘𝑆))
7	6	necomd 3013	. . . . . 6 ⊢ ((𝑁‘𝑇) ⊊ (𝑁‘𝑆) → (𝑁‘𝑆) ≠ (𝑁‘𝑇))
8	7	necon2bi 2988	. . . . 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 17669	. . . . . . 7 ⊢ (𝜑 → 𝑆 ⊆ 𝑋)
13	1, 2, 11, 12	mrieqv2d 17672	. . . . . 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 5281	. . . . . 6 ⊢ (𝜑 → 𝑇 ∈ V)
16		simpr 488	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 = 𝑇) → 𝑠 = 𝑇)
17	16	psseq1d 4049	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 = 𝑇) → (𝑠 ⊊ 𝑆 ↔ 𝑇 ⊊ 𝑆))
18	16	fveq2d 6872	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 = 𝑇) → (𝑁‘𝑠) = (𝑁‘𝑇))
19	18	psseq1d 4049	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 = 𝑇) → ((𝑁‘𝑠) ⊊ (𝑁‘𝑆) ↔ (𝑁‘𝑇) ⊊ (𝑁‘𝑆)))
20	17, 19	imbi12d 346	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 = 𝑇) → ((𝑠 ⊊ 𝑆 → (𝑁‘𝑠) ⊊ (𝑁‘𝑆)) ↔ (𝑇 ⊊ 𝑆 → (𝑁‘𝑇) ⊊ (𝑁‘𝑆))))
21	15, 20	spcdv 3554	. . . . 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 4056	. . . . 5 ⊢ (𝑇 ⊆ 𝑆 ↔ (𝑇 ⊊ 𝑆 ∨ 𝑇 = 𝑆))
25	4, 24	sylib 220	. . . 4 ⊢ (𝜑 → (𝑇 ⊊ 𝑆 ∨ 𝑇 = 𝑆))
26	25	ord 875	. . 3 ⊢ (𝜑 → (¬ 𝑇 ⊊ 𝑆 → 𝑇 = 𝑆))
27	23, 26	mpd 15	. 2 ⊢ (𝜑 → 𝑇 = 𝑆)
28	27	eqcomd 2769	1 ⊢ (𝜑 → 𝑆 = 𝑇)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∨ wo 858  ∀wal 1559   = wceq 1561   ∈ wcel 2143  Vcvv 3455   ⊆ wss 3905   ⊊ wpss 3906  ‘cfv 6522  Moorecmre 17611  mrClscmrc 17612  mrIndcmri 17613

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5247  ax-nul 5257  ax-pow 5323  ax-pr 5391  ax-un 7719

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3078  df-rex 3088  df-rab 3416  df-v 3457  df-sbc 3746  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-br 5102  df-opab 5164  df-mpt 5183  df-id 5543  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-iota 6478  df-fun 6524  df-fn 6525  df-f 6526  df-fv 6530  df-mre 17615  df-mrc 17616  df-mri 17617

This theorem is referenced by:  mreexexlem3d  17679  acsmap2d  18588