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Theorem ismri2dad 16900
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 16899 . . . 4 (𝜑𝑆𝑋)
62, 3, 4, 5ismri2d 16896 . . 3 (𝜑 → (𝑆𝐼 ↔ ∀𝑥𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))))
71, 6mpbid 234 . 2 (𝜑 → ∀𝑥𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})))
8 ismri2dad.5 . . 3 (𝜑𝑌𝑆)
9 simpr 487 . . . . 5 ((𝜑𝑥 = 𝑌) → 𝑥 = 𝑌)
109sneqd 4571 . . . . . . 7 ((𝜑𝑥 = 𝑌) → {𝑥} = {𝑌})
1110difeq2d 4097 . . . . . 6 ((𝜑𝑥 = 𝑌) → (𝑆 ∖ {𝑥}) = (𝑆 ∖ {𝑌}))
1211fveq2d 6667 . . . . 5 ((𝜑𝑥 = 𝑌) → (𝑁‘(𝑆 ∖ {𝑥})) = (𝑁‘(𝑆 ∖ {𝑌})))
139, 12eleq12d 2905 . . . 4 ((𝜑𝑥 = 𝑌) → (𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})) ↔ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))))
1413notbid 320 . . 3 ((𝜑𝑥 = 𝑌) → (¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})) ↔ ¬ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))))
158, 14rspcdv 3613 . 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 398   = wceq 1531  wcel 2108  wral 3136  cdif 3931  {csn 4559  cfv 6348  Moorecmre 16845  mrClscmrc 16846  mrIndcmri 16847
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-iota 6307  df-fun 6350  df-fv 6356  df-mre 16849  df-mri 16851
This theorem is referenced by:  mrieqv2d  16902  mreexmrid  16906  mreexexlem2d  16908  acsfiindd  17779
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