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Mirrors > Home > MPE Home > Th. List > ismri2dad | Structured version Visualization version GIF version |
Description: Consequence of a set in a Moore system being independent. Deduction form. (Contributed by David Moews, 1-May-2017.) |
Ref | Expression |
---|---|
ismri2dad.1 | ⊢ 𝑁 = (mrCls‘𝐴) |
ismri2dad.2 | ⊢ 𝐼 = (mrInd‘𝐴) |
ismri2dad.3 | ⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋)) |
ismri2dad.4 | ⊢ (𝜑 → 𝑆 ∈ 𝐼) |
ismri2dad.5 | ⊢ (𝜑 → 𝑌 ∈ 𝑆) |
Ref | Expression |
---|---|
ismri2dad | ⊢ (𝜑 → ¬ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ismri2dad.4 | . . 3 ⊢ (𝜑 → 𝑆 ∈ 𝐼) | |
2 | ismri2dad.1 | . . . 4 ⊢ 𝑁 = (mrCls‘𝐴) | |
3 | ismri2dad.2 | . . . 4 ⊢ 𝐼 = (mrInd‘𝐴) | |
4 | ismri2dad.3 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋)) | |
5 | 3, 4, 1 | mrissd 17163 | . . . 4 ⊢ (𝜑 → 𝑆 ⊆ 𝑋) |
6 | 2, 3, 4, 5 | ismri2d 17160 | . . 3 ⊢ (𝜑 → (𝑆 ∈ 𝐼 ↔ ∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})))) |
7 | 1, 6 | mpbid 235 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))) |
8 | ismri2dad.5 | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑆) | |
9 | simpr 488 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑌) → 𝑥 = 𝑌) | |
10 | 9 | sneqd 4567 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝑌) → {𝑥} = {𝑌}) |
11 | 10 | difeq2d 4051 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑌) → (𝑆 ∖ {𝑥}) = (𝑆 ∖ {𝑌})) |
12 | 11 | fveq2d 6739 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑌) → (𝑁‘(𝑆 ∖ {𝑥})) = (𝑁‘(𝑆 ∖ {𝑌}))) |
13 | 9, 12 | eleq12d 2833 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑌) → (𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})) ↔ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌})))) |
14 | 13 | notbid 321 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑌) → (¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})) ↔ ¬ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌})))) |
15 | 8, 14 | rspcdv 3541 | . 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})) → ¬ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌})))) |
16 | 7, 15 | mpd 15 | 1 ⊢ (𝜑 → ¬ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2111 ∀wral 3062 ∖ cdif 3877 {csn 4555 ‘cfv 6397 Moorecmre 17109 mrClscmrc 17110 mrIndcmri 17111 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-sep 5206 ax-nul 5213 ax-pow 5272 ax-pr 5336 ax-un 7541 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ne 2942 df-ral 3067 df-rex 3068 df-rab 3071 df-v 3422 df-dif 3883 df-un 3885 df-in 3887 df-ss 3897 df-nul 4252 df-if 4454 df-pw 4529 df-sn 4556 df-pr 4558 df-op 4562 df-uni 4834 df-br 5068 df-opab 5130 df-mpt 5150 df-id 5469 df-xp 5571 df-rel 5572 df-cnv 5573 df-co 5574 df-dm 5575 df-rn 5576 df-iota 6355 df-fun 6399 df-fv 6405 df-mre 17113 df-mri 17115 |
This theorem is referenced by: mrieqv2d 17166 mreexmrid 17170 mreexexlem2d 17172 acsfiindd 18083 |
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