Theorem mrissmrcd 17546

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 17533, and so are equal by mrieqv2d 17545.) (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 17533	. . . . 5 ⊢ (𝜑 → (𝑁‘𝑆) = (𝑁‘𝑇))
6		pssne 4046	. . . . . . 7 ⊢ ((𝑁‘𝑇) ⊊ (𝑁‘𝑆) → (𝑁‘𝑇) ≠ (𝑁‘𝑆))
7	6	necomd 2983	. . . . . 6 ⊢ ((𝑁‘𝑇) ⊊ (𝑁‘𝑆) → (𝑁‘𝑆) ≠ (𝑁‘𝑇))
8	7	necon2bi 2958	. . . . 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 17542	. . . . . . 7 ⊢ (𝜑 → 𝑆 ⊆ 𝑋)
13	1, 2, 11, 12	mrieqv2d 17545	. . . . . 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 5260	. . . . . 6 ⊢ (𝜑 → 𝑇 ∈ V)
16		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 = 𝑇) → 𝑠 = 𝑇)
17	16	psseq1d 4042	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 = 𝑇) → (𝑠 ⊊ 𝑆 ↔ 𝑇 ⊊ 𝑆))
18	16	fveq2d 6826	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 = 𝑇) → (𝑁‘𝑠) = (𝑁‘𝑇))
19	18	psseq1d 4042	. . . . . . 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 3544	. . . . 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 4049	. . . . 5 ⊢ (𝑇 ⊆ 𝑆 ↔ (𝑇 ⊊ 𝑆 ∨ 𝑇 = 𝑆))
25	4, 24	sylib 218	. . . 4 ⊢ (𝜑 → (𝑇 ⊊ 𝑆 ∨ 𝑇 = 𝑆))
26	25	ord 864	. . 3 ⊢ (𝜑 → (¬ 𝑇 ⊊ 𝑆 → 𝑇 = 𝑆))
27	23, 26	mpd 15	. 2 ⊢ (𝜑 → 𝑇 = 𝑆)
28	27	eqcomd 2737	1 ⊢ (𝜑 → 𝑆 = 𝑇)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 847  ∀wal 1539   = wceq 1541   ∈ wcel 2111  Vcvv 3436   ⊆ wss 3897   ⊊ wpss 3898  ‘cfv 6481  Moorecmre 17484  mrClscmrc 17485  mrIndcmri 17486

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 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-int 4896  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-fv 6489  df-mre 17488  df-mrc 17489  df-mri 17490

This theorem is referenced by:  mreexexlem3d  17552  acsmap2d  18461