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Theorem mrissmrcd 17584
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 17571, and so are equal by mrieqv2d 17583.) (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 17571 . . . . 5 (πœ‘ β†’ (π‘β€˜π‘†) = (π‘β€˜π‘‡))
6 pssne 4097 . . . . . . 7 ((π‘β€˜π‘‡) ⊊ (π‘β€˜π‘†) β†’ (π‘β€˜π‘‡) β‰  (π‘β€˜π‘†))
76necomd 2997 . . . . . 6 ((π‘β€˜π‘‡) ⊊ (π‘β€˜π‘†) β†’ (π‘β€˜π‘†) β‰  (π‘β€˜π‘‡))
87necon2bi 2972 . . . . 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 17580 . . . . . . 7 (πœ‘ β†’ 𝑆 βŠ† 𝑋)
131, 2, 11, 12mrieqv2d 17583 . . . . . 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 5325 . . . . . 6 (πœ‘ β†’ 𝑇 ∈ V)
16 simpr 486 . . . . . . . 8 ((πœ‘ ∧ 𝑠 = 𝑇) β†’ 𝑠 = 𝑇)
1716psseq1d 4093 . . . . . . 7 ((πœ‘ ∧ 𝑠 = 𝑇) β†’ (𝑠 ⊊ 𝑆 ↔ 𝑇 ⊊ 𝑆))
1816fveq2d 6896 . . . . . . . 8 ((πœ‘ ∧ 𝑠 = 𝑇) β†’ (π‘β€˜π‘ ) = (π‘β€˜π‘‡))
1918psseq1d 4093 . . . . . . 7 ((πœ‘ ∧ 𝑠 = 𝑇) β†’ ((π‘β€˜π‘ ) ⊊ (π‘β€˜π‘†) ↔ (π‘β€˜π‘‡) ⊊ (π‘β€˜π‘†)))
2017, 19imbi12d 345 . . . . . 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 3585 . . . . 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 4100 . . . . 5 (𝑇 βŠ† 𝑆 ↔ (𝑇 ⊊ 𝑆 ∨ 𝑇 = 𝑆))
254, 24sylib 217 . . . 4 (πœ‘ β†’ (𝑇 ⊊ 𝑆 ∨ 𝑇 = 𝑆))
2625ord 863 . . 3 (πœ‘ β†’ (Β¬ 𝑇 ⊊ 𝑆 β†’ 𝑇 = 𝑆))
2723, 26mpd 15 . 2 (πœ‘ β†’ 𝑇 = 𝑆)
2827eqcomd 2739 1 (πœ‘ β†’ 𝑆 = 𝑇)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 397   ∨ wo 846  βˆ€wal 1540   = wceq 1542   ∈ wcel 2107  Vcvv 3475   βŠ† wss 3949   ⊊ wpss 3950  β€˜cfv 6544  Moorecmre 17526  mrClscmrc 17527  mrIndcmri 17528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fv 6552  df-mre 17530  df-mrc 17531  df-mri 17532
This theorem is referenced by:  mreexexlem3d  17590  acsmap2d  18508
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