Theorem mrissmrcd 17581

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 17568, and so are equal by mrieqv2d 17580.) (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 17568	. . . . 5 ⊢ (𝜑 → (𝑁‘𝑆) = (𝑁‘𝑇))
6		pssne 4058	. . . . . . 7 ⊢ ((𝑁‘𝑇) ⊊ (𝑁‘𝑆) → (𝑁‘𝑇) ≠ (𝑁‘𝑆))
7	6	necomd 2980	. . . . . 6 ⊢ ((𝑁‘𝑇) ⊊ (𝑁‘𝑆) → (𝑁‘𝑆) ≠ (𝑁‘𝑇))
8	7	necon2bi 2955	. . . . 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 17577	. . . . . . 7 ⊢ (𝜑 → 𝑆 ⊆ 𝑋)
13	1, 2, 11, 12	mrieqv2d 17580	. . . . . 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 5274	. . . . . 6 ⊢ (𝜑 → 𝑇 ∈ V)
16		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 = 𝑇) → 𝑠 = 𝑇)
17	16	psseq1d 4054	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 = 𝑇) → (𝑠 ⊊ 𝑆 ↔ 𝑇 ⊊ 𝑆))
18	16	fveq2d 6844	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 = 𝑇) → (𝑁‘𝑠) = (𝑁‘𝑇))
19	18	psseq1d 4054	. . . . . . 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 3557	. . . . 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 4061	. . . . 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 2735	1 ⊢ (𝜑 → 𝑆 = 𝑇)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 847  ∀wal 1538   = wceq 1540   ∈ wcel 2109  Vcvv 3444   ⊆ wss 3911   ⊊ wpss 3912  ‘cfv 6499  Moorecmre 17519  mrClscmrc 17520  mrIndcmri 17521

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-int 4907  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-fv 6507  df-mre 17523  df-mrc 17524  df-mri 17525

This theorem is referenced by:  mreexexlem3d  17587  acsmap2d  18496