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Theorem ismri2dad 16650
Description: Consequence of a set in a Moore system being independent. Deduction form. (Contributed by David Moews, 1-May-2017.)
Hypotheses
Ref Expression
ismri2dad.1 𝑁 = (mrCls‘𝐴)
ismri2dad.2 𝐼 = (mrInd‘𝐴)
ismri2dad.3 (𝜑𝐴 ∈ (Moore‘𝑋))
ismri2dad.4 (𝜑𝑆𝐼)
ismri2dad.5 (𝜑𝑌𝑆)
Assertion
Ref Expression
ismri2dad (𝜑 → ¬ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌})))

Proof of Theorem ismri2dad
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 ismri2dad.4 . . 3 (𝜑𝑆𝐼)
2 ismri2dad.1 . . . 4 𝑁 = (mrCls‘𝐴)
3 ismri2dad.2 . . . 4 𝐼 = (mrInd‘𝐴)
4 ismri2dad.3 . . . 4 (𝜑𝐴 ∈ (Moore‘𝑋))
53, 4, 1mrissd 16649 . . . 4 (𝜑𝑆𝑋)
62, 3, 4, 5ismri2d 16646 . . 3 (𝜑 → (𝑆𝐼 ↔ ∀𝑥𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))))
71, 6mpbid 224 . 2 (𝜑 → ∀𝑥𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})))
8 ismri2dad.5 . . 3 (𝜑𝑌𝑆)
9 simpr 479 . . . . 5 ((𝜑𝑥 = 𝑌) → 𝑥 = 𝑌)
109sneqd 4409 . . . . . . 7 ((𝜑𝑥 = 𝑌) → {𝑥} = {𝑌})
1110difeq2d 3955 . . . . . 6 ((𝜑𝑥 = 𝑌) → (𝑆 ∖ {𝑥}) = (𝑆 ∖ {𝑌}))
1211fveq2d 6437 . . . . 5 ((𝜑𝑥 = 𝑌) → (𝑁‘(𝑆 ∖ {𝑥})) = (𝑁‘(𝑆 ∖ {𝑌})))
139, 12eleq12d 2900 . . . 4 ((𝜑𝑥 = 𝑌) → (𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})) ↔ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))))
1413notbid 310 . . 3 ((𝜑𝑥 = 𝑌) → (¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})) ↔ ¬ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))))
158, 14rspcdv 3529 . 2 (𝜑 → (∀𝑥𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})) → ¬ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))))
167, 15mpd 15 1 (𝜑 → ¬ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌})))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 386   = wceq 1656  wcel 2164  wral 3117  cdif 3795  {csn 4397  cfv 6123  Moorecmre 16595  mrClscmrc 16596  mrIndcmri 16597
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-iota 6086  df-fun 6125  df-fv 6131  df-mre 16599  df-mri 16601
This theorem is referenced by:  mrieqv2d  16652  mreexmrid  16656  mreexexlem2d  16658  acsfiindd  17530
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