Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  mrissmrcd Structured version   Visualization version   GIF version

Theorem mrissmrcd 16221
 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 16208, and so are equal by mrieqv2d 16220.) (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 16208 . . . . 5 (𝜑 → (𝑁𝑆) = (𝑁𝑇))
6 pssne 3681 . . . . . . 7 ((𝑁𝑇) ⊊ (𝑁𝑆) → (𝑁𝑇) ≠ (𝑁𝑆))
76necomd 2845 . . . . . 6 ((𝑁𝑇) ⊊ (𝑁𝑆) → (𝑁𝑆) ≠ (𝑁𝑇))
87necon2bi 2820 . . . . 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 16217 . . . . . . 7 (𝜑𝑆𝑋)
131, 2, 11, 12mrieqv2d 16220 . . . . . 6 (𝜑 → (𝑆𝐼 ↔ ∀𝑠(𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆))))
1410, 13mpbid 222 . . . . 5 (𝜑 → ∀𝑠(𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆)))
1510, 4ssexd 4765 . . . . . 6 (𝜑𝑇 ∈ V)
16 simpr 477 . . . . . . . 8 ((𝜑𝑠 = 𝑇) → 𝑠 = 𝑇)
1716psseq1d 3677 . . . . . . 7 ((𝜑𝑠 = 𝑇) → (𝑠𝑆𝑇𝑆))
1816fveq2d 6152 . . . . . . . 8 ((𝜑𝑠 = 𝑇) → (𝑁𝑠) = (𝑁𝑇))
1918psseq1d 3677 . . . . . . 7 ((𝜑𝑠 = 𝑇) → ((𝑁𝑠) ⊊ (𝑁𝑆) ↔ (𝑁𝑇) ⊊ (𝑁𝑆)))
2017, 19imbi12d 334 . . . . . 6 ((𝜑𝑠 = 𝑇) → ((𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆)) ↔ (𝑇𝑆 → (𝑁𝑇) ⊊ (𝑁𝑆))))
2115, 20spcdv 3277 . . . . 5 (𝜑 → (∀𝑠(𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆)) → (𝑇𝑆 → (𝑁𝑇) ⊊ (𝑁𝑆))))
2214, 21mpd 15 . . . 4 (𝜑 → (𝑇𝑆 → (𝑁𝑇) ⊊ (𝑁𝑆)))
239, 22mtod 189 . . 3 (𝜑 → ¬ 𝑇𝑆)
24 sspss 3684 . . . . 5 (𝑇𝑆 ↔ (𝑇𝑆𝑇 = 𝑆))
254, 24sylib 208 . . . 4 (𝜑 → (𝑇𝑆𝑇 = 𝑆))
2625ord 392 . . 3 (𝜑 → (¬ 𝑇𝑆𝑇 = 𝑆))
2723, 26mpd 15 . 2 (𝜑𝑇 = 𝑆)
2827eqcomd 2627 1 (𝜑𝑆 = 𝑇)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 383   ∧ wa 384  ∀wal 1478   = wceq 1480   ∈ wcel 1987  Vcvv 3186   ⊆ wss 3555   ⊊ wpss 3556  ‘cfv 5847  Moorecmre 16163  mrClscmrc 16164  mrIndcmri 16165 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-int 4441  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fv 5855  df-mre 16167  df-mrc 16168  df-mri 16169 This theorem is referenced by:  mreexexlem3d  16227  acsmap2d  17100
 Copyright terms: Public domain W3C validator