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Theorem mrissmrcd 17685
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 17672, and so are equal by mrieqv2d 17684.) (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 17672 . . . . 5 (𝜑 → (𝑁𝑆) = (𝑁𝑇))
6 pssne 4109 . . . . . . 7 ((𝑁𝑇) ⊊ (𝑁𝑆) → (𝑁𝑇) ≠ (𝑁𝑆))
76necomd 2994 . . . . . 6 ((𝑁𝑇) ⊊ (𝑁𝑆) → (𝑁𝑆) ≠ (𝑁𝑇))
87necon2bi 2969 . . . . 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 17681 . . . . . . 7 (𝜑𝑆𝑋)
131, 2, 11, 12mrieqv2d 17684 . . . . . 6 (𝜑 → (𝑆𝐼 ↔ ∀𝑠(𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆))))
1410, 13mpbid 232 . . . . 5 (𝜑 → ∀𝑠(𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆)))
1510, 4ssexd 5330 . . . . . 6 (𝜑𝑇 ∈ V)
16 simpr 484 . . . . . . . 8 ((𝜑𝑠 = 𝑇) → 𝑠 = 𝑇)
1716psseq1d 4105 . . . . . . 7 ((𝜑𝑠 = 𝑇) → (𝑠𝑆𝑇𝑆))
1816fveq2d 6911 . . . . . . . 8 ((𝜑𝑠 = 𝑇) → (𝑁𝑠) = (𝑁𝑇))
1918psseq1d 4105 . . . . . . 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 4112 . . . . 5 (𝑇𝑆 ↔ (𝑇𝑆𝑇 = 𝑆))
254, 24sylib 218 . . . 4 (𝜑 → (𝑇𝑆𝑇 = 𝑆))
2625ord 864 . . 3 (𝜑 → (¬ 𝑇𝑆𝑇 = 𝑆))
2723, 26mpd 15 . 2 (𝜑𝑇 = 𝑆)
2827eqcomd 2741 1 (𝜑𝑆 = 𝑇)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wo 847  wal 1535   = wceq 1537  wcel 2106  Vcvv 3478  wss 3963  wpss 3964  cfv 6563  Moorecmre 17627  mrClscmrc 17628  mrIndcmri 17629
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-mre 17631  df-mrc 17632  df-mri 17633
This theorem is referenced by:  mreexexlem3d  17691  acsmap2d  18613
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