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Theorem mrissmrcd 17683
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 17670, and so are equal by mrieqv2d 17682.) (Contributed by David Moews, 1-May-2017.)
Hypotheses
Ref Expression
mrissmrcd.1 (𝜑𝐴 ∈ (Moore‘𝑋))
mrissmrcd.2 𝑁 = (mrCls‘𝐴)
mrissmrcd.3 𝐼 = (mrInd‘𝐴)
mrissmrcd.4 (𝜑𝑆 ⊆ (𝑁𝑇))
mrissmrcd.5 (𝜑𝑇𝑆)
mrissmrcd.6 (𝜑𝑆𝐼)
Assertion
Ref Expression
mrissmrcd (𝜑𝑆 = 𝑇)

Proof of Theorem mrissmrcd
Dummy variable 𝑠 is distinct from all other variables.
StepHypRef Expression
1 mrissmrcd.1 . . . . . 6 (𝜑𝐴 ∈ (Moore‘𝑋))
2 mrissmrcd.2 . . . . . 6 𝑁 = (mrCls‘𝐴)
3 mrissmrcd.4 . . . . . 6 (𝜑𝑆 ⊆ (𝑁𝑇))
4 mrissmrcd.5 . . . . . 6 (𝜑𝑇𝑆)
51, 2, 3, 4mressmrcd 17670 . . . . 5 (𝜑 → (𝑁𝑆) = (𝑁𝑇))
6 pssne 4099 . . . . . . 7 ((𝑁𝑇) ⊊ (𝑁𝑆) → (𝑁𝑇) ≠ (𝑁𝑆))
76necomd 2996 . . . . . 6 ((𝑁𝑇) ⊊ (𝑁𝑆) → (𝑁𝑆) ≠ (𝑁𝑇))
87necon2bi 2971 . . . . 5 ((𝑁𝑆) = (𝑁𝑇) → ¬ (𝑁𝑇) ⊊ (𝑁𝑆))
95, 8syl 17 . . . 4 (𝜑 → ¬ (𝑁𝑇) ⊊ (𝑁𝑆))
10 mrissmrcd.6 . . . . . 6 (𝜑𝑆𝐼)
11 mrissmrcd.3 . . . . . . 7 𝐼 = (mrInd‘𝐴)
1211, 1, 10mrissd 17679 . . . . . . 7 (𝜑𝑆𝑋)
131, 2, 11, 12mrieqv2d 17682 . . . . . 6 (𝜑 → (𝑆𝐼 ↔ ∀𝑠(𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆))))
1410, 13mpbid 232 . . . . 5 (𝜑 → ∀𝑠(𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆)))
1510, 4ssexd 5324 . . . . . 6 (𝜑𝑇 ∈ V)
16 simpr 484 . . . . . . . 8 ((𝜑𝑠 = 𝑇) → 𝑠 = 𝑇)
1716psseq1d 4095 . . . . . . 7 ((𝜑𝑠 = 𝑇) → (𝑠𝑆𝑇𝑆))
1816fveq2d 6910 . . . . . . . 8 ((𝜑𝑠 = 𝑇) → (𝑁𝑠) = (𝑁𝑇))
1918psseq1d 4095 . . . . . . 7 ((𝜑𝑠 = 𝑇) → ((𝑁𝑠) ⊊ (𝑁𝑆) ↔ (𝑁𝑇) ⊊ (𝑁𝑆)))
2017, 19imbi12d 344 . . . . . 6 ((𝜑𝑠 = 𝑇) → ((𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆)) ↔ (𝑇𝑆 → (𝑁𝑇) ⊊ (𝑁𝑆))))
2115, 20spcdv 3594 . . . . 5 (𝜑 → (∀𝑠(𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆)) → (𝑇𝑆 → (𝑁𝑇) ⊊ (𝑁𝑆))))
2214, 21mpd 15 . . . 4 (𝜑 → (𝑇𝑆 → (𝑁𝑇) ⊊ (𝑁𝑆)))
239, 22mtod 198 . . 3 (𝜑 → ¬ 𝑇𝑆)
24 sspss 4102 . . . . 5 (𝑇𝑆 ↔ (𝑇𝑆𝑇 = 𝑆))
254, 24sylib 218 . . . 4 (𝜑 → (𝑇𝑆𝑇 = 𝑆))
2625ord 865 . . 3 (𝜑 → (¬ 𝑇𝑆𝑇 = 𝑆))
2723, 26mpd 15 . 2 (𝜑𝑇 = 𝑆)
2827eqcomd 2743 1 (𝜑𝑆 = 𝑇)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wo 848  wal 1538   = wceq 1540  wcel 2108  Vcvv 3480  wss 3951  wpss 3952  cfv 6561  Moorecmre 17625  mrClscmrc 17626  mrIndcmri 17627
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-mre 17629  df-mrc 17630  df-mri 17631
This theorem is referenced by:  mreexexlem3d  17689  acsmap2d  18600
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