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Theorem ismri2dad 16218
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 16217 . . . 4 (𝜑𝑆𝑋)
62, 3, 4, 5ismri2d 16214 . . 3 (𝜑 → (𝑆𝐼 ↔ ∀𝑥𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))))
71, 6mpbid 222 . 2 (𝜑 → ∀𝑥𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})))
8 ismri2dad.5 . . 3 (𝜑𝑌𝑆)
9 simpr 477 . . . . 5 ((𝜑𝑥 = 𝑌) → 𝑥 = 𝑌)
109sneqd 4160 . . . . . . 7 ((𝜑𝑥 = 𝑌) → {𝑥} = {𝑌})
1110difeq2d 3706 . . . . . 6 ((𝜑𝑥 = 𝑌) → (𝑆 ∖ {𝑥}) = (𝑆 ∖ {𝑌}))
1211fveq2d 6152 . . . . 5 ((𝜑𝑥 = 𝑌) → (𝑁‘(𝑆 ∖ {𝑥})) = (𝑁‘(𝑆 ∖ {𝑌})))
139, 12eleq12d 2692 . . . 4 ((𝜑𝑥 = 𝑌) → (𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})) ↔ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))))
1413notbid 308 . . 3 ((𝜑𝑥 = 𝑌) → (¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})) ↔ ¬ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))))
158, 14rspcdv 3298 . 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 384   = wceq 1480  wcel 1987  wral 2907  cdif 3552  {csn 4148  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-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  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-iota 5810  df-fun 5849  df-fv 5855  df-mre 16167  df-mri 16169
This theorem is referenced by:  mrieqv2d  16220  mreexmrid  16224  mreexexlem2d  16226  acsfiindd  17098
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