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Theorem mrissmrcd 17586
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 17573, and so are equal by mrieqv2d 17585.) (Contributed by David Moews, 1-May-2017.)
Hypotheses
Ref Expression
mrissmrcd.1 (πœ‘ β†’ 𝐴 ∈ (Mooreβ€˜π‘‹))
mrissmrcd.2 𝑁 = (mrClsβ€˜π΄)
mrissmrcd.3 𝐼 = (mrIndβ€˜π΄)
mrissmrcd.4 (πœ‘ β†’ 𝑆 βŠ† (π‘β€˜π‘‡))
mrissmrcd.5 (πœ‘ β†’ 𝑇 βŠ† 𝑆)
mrissmrcd.6 (πœ‘ β†’ 𝑆 ∈ 𝐼)
Assertion
Ref Expression
mrissmrcd (πœ‘ β†’ 𝑆 = 𝑇)

Proof of Theorem mrissmrcd
Dummy variable 𝑠 is distinct from all other variables.
StepHypRef Expression
1 mrissmrcd.1 . . . . . 6 (πœ‘ β†’ 𝐴 ∈ (Mooreβ€˜π‘‹))
2 mrissmrcd.2 . . . . . 6 𝑁 = (mrClsβ€˜π΄)
3 mrissmrcd.4 . . . . . 6 (πœ‘ β†’ 𝑆 βŠ† (π‘β€˜π‘‡))
4 mrissmrcd.5 . . . . . 6 (πœ‘ β†’ 𝑇 βŠ† 𝑆)
51, 2, 3, 4mressmrcd 17573 . . . . 5 (πœ‘ β†’ (π‘β€˜π‘†) = (π‘β€˜π‘‡))
6 pssne 4096 . . . . . . 7 ((π‘β€˜π‘‡) ⊊ (π‘β€˜π‘†) β†’ (π‘β€˜π‘‡) β‰  (π‘β€˜π‘†))
76necomd 2996 . . . . . 6 ((π‘β€˜π‘‡) ⊊ (π‘β€˜π‘†) β†’ (π‘β€˜π‘†) β‰  (π‘β€˜π‘‡))
87necon2bi 2971 . . . . 5 ((π‘β€˜π‘†) = (π‘β€˜π‘‡) β†’ Β¬ (π‘β€˜π‘‡) ⊊ (π‘β€˜π‘†))
95, 8syl 17 . . . 4 (πœ‘ β†’ Β¬ (π‘β€˜π‘‡) ⊊ (π‘β€˜π‘†))
10 mrissmrcd.6 . . . . . 6 (πœ‘ β†’ 𝑆 ∈ 𝐼)
11 mrissmrcd.3 . . . . . . 7 𝐼 = (mrIndβ€˜π΄)
1211, 1, 10mrissd 17582 . . . . . . 7 (πœ‘ β†’ 𝑆 βŠ† 𝑋)
131, 2, 11, 12mrieqv2d 17585 . . . . . 6 (πœ‘ β†’ (𝑆 ∈ 𝐼 ↔ βˆ€π‘ (𝑠 ⊊ 𝑆 β†’ (π‘β€˜π‘ ) ⊊ (π‘β€˜π‘†))))
1410, 13mpbid 231 . . . . 5 (πœ‘ β†’ βˆ€π‘ (𝑠 ⊊ 𝑆 β†’ (π‘β€˜π‘ ) ⊊ (π‘β€˜π‘†)))
1510, 4ssexd 5324 . . . . . 6 (πœ‘ β†’ 𝑇 ∈ V)
16 simpr 485 . . . . . . . 8 ((πœ‘ ∧ 𝑠 = 𝑇) β†’ 𝑠 = 𝑇)
1716psseq1d 4092 . . . . . . 7 ((πœ‘ ∧ 𝑠 = 𝑇) β†’ (𝑠 ⊊ 𝑆 ↔ 𝑇 ⊊ 𝑆))
1816fveq2d 6895 . . . . . . . 8 ((πœ‘ ∧ 𝑠 = 𝑇) β†’ (π‘β€˜π‘ ) = (π‘β€˜π‘‡))
1918psseq1d 4092 . . . . . . 7 ((πœ‘ ∧ 𝑠 = 𝑇) β†’ ((π‘β€˜π‘ ) ⊊ (π‘β€˜π‘†) ↔ (π‘β€˜π‘‡) ⊊ (π‘β€˜π‘†)))
2017, 19imbi12d 344 . . . . . 6 ((πœ‘ ∧ 𝑠 = 𝑇) β†’ ((𝑠 ⊊ 𝑆 β†’ (π‘β€˜π‘ ) ⊊ (π‘β€˜π‘†)) ↔ (𝑇 ⊊ 𝑆 β†’ (π‘β€˜π‘‡) ⊊ (π‘β€˜π‘†))))
2115, 20spcdv 3584 . . . . 5 (πœ‘ β†’ (βˆ€π‘ (𝑠 ⊊ 𝑆 β†’ (π‘β€˜π‘ ) ⊊ (π‘β€˜π‘†)) β†’ (𝑇 ⊊ 𝑆 β†’ (π‘β€˜π‘‡) ⊊ (π‘β€˜π‘†))))
2214, 21mpd 15 . . . 4 (πœ‘ β†’ (𝑇 ⊊ 𝑆 β†’ (π‘β€˜π‘‡) ⊊ (π‘β€˜π‘†)))
239, 22mtod 197 . . 3 (πœ‘ β†’ Β¬ 𝑇 ⊊ 𝑆)
24 sspss 4099 . . . . 5 (𝑇 βŠ† 𝑆 ↔ (𝑇 ⊊ 𝑆 ∨ 𝑇 = 𝑆))
254, 24sylib 217 . . . 4 (πœ‘ β†’ (𝑇 ⊊ 𝑆 ∨ 𝑇 = 𝑆))
2625ord 862 . . 3 (πœ‘ β†’ (Β¬ 𝑇 ⊊ 𝑆 β†’ 𝑇 = 𝑆))
2723, 26mpd 15 . 2 (πœ‘ β†’ 𝑇 = 𝑆)
2827eqcomd 2738 1 (πœ‘ β†’ 𝑆 = 𝑇)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 396   ∨ wo 845  βˆ€wal 1539   = wceq 1541   ∈ wcel 2106  Vcvv 3474   βŠ† wss 3948   ⊊ wpss 3949  β€˜cfv 6543  Moorecmre 17528  mrClscmrc 17529  mrIndcmri 17530
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-mre 17532  df-mrc 17533  df-mri 17534
This theorem is referenced by:  mreexexlem3d  17592  acsmap2d  18510
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