Theorem mrissmrcd 17698

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 17685, and so are equal by mrieqv2d 17697.) (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 17685	. . . . 5 ⊢ (𝜑 → (𝑁‘𝑆) = (𝑁‘𝑇))
6		pssne 4122	. . . . . . 7 ⊢ ((𝑁‘𝑇) ⊊ (𝑁‘𝑆) → (𝑁‘𝑇) ≠ (𝑁‘𝑆))
7	6	necomd 3002	. . . . . 6 ⊢ ((𝑁‘𝑇) ⊊ (𝑁‘𝑆) → (𝑁‘𝑆) ≠ (𝑁‘𝑇))
8	7	necon2bi 2977	. . . . 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 17694	. . . . . . 7 ⊢ (𝜑 → 𝑆 ⊆ 𝑋)
13	1, 2, 11, 12	mrieqv2d 17697	. . . . . 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 5342	. . . . . 6 ⊢ (𝜑 → 𝑇 ∈ V)
16		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 = 𝑇) → 𝑠 = 𝑇)
17	16	psseq1d 4118	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 = 𝑇) → (𝑠 ⊊ 𝑆 ↔ 𝑇 ⊊ 𝑆))
18	16	fveq2d 6924	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 = 𝑇) → (𝑁‘𝑠) = (𝑁‘𝑇))
19	18	psseq1d 4118	. . . . . . 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 3607	. . . . 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 4125	. . . . 5 ⊢ (𝑇 ⊆ 𝑆 ↔ (𝑇 ⊊ 𝑆 ∨ 𝑇 = 𝑆))
25	4, 24	sylib 218	. . . 4 ⊢ (𝜑 → (𝑇 ⊊ 𝑆 ∨ 𝑇 = 𝑆))
26	25	ord 863	. . 3 ⊢ (𝜑 → (¬ 𝑇 ⊊ 𝑆 → 𝑇 = 𝑆))
27	23, 26	mpd 15	. 2 ⊢ (𝜑 → 𝑇 = 𝑆)
28	27	eqcomd 2746	1 ⊢ (𝜑 → 𝑆 = 𝑇)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 846  ∀wal 1535   = wceq 1537   ∈ wcel 2108  Vcvv 3488   ⊆ wss 3976   ⊊ wpss 3977  ‘cfv 6573  Moorecmre 17640  mrClscmrc 17641  mrIndcmri 17642

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-fv 6581  df-mre 17644  df-mrc 17645  df-mri 17646

This theorem is referenced by:  mreexexlem3d  17704  acsmap2d  18625