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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 16899 | . . . 4 ⊢ (𝜑 → 𝑆 ⊆ 𝑋) |
6 | 2, 3, 4, 5 | ismri2d 16896 | . . 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 4537 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝑌) → {𝑥} = {𝑌}) |
11 | 10 | difeq2d 4050 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑌) → (𝑆 ∖ {𝑥}) = (𝑆 ∖ {𝑌})) |
12 | 11 | fveq2d 6649 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑌) → (𝑁‘(𝑆 ∖ {𝑥})) = (𝑁‘(𝑆 ∖ {𝑌}))) |
13 | 9, 12 | eleq12d 2884 | . . . 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 3563 | . 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 1538 ∈ wcel 2111 ∀wral 3106 ∖ cdif 3878 {csn 4525 ‘cfv 6324 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 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-iota 6283 df-fun 6326 df-fv 6332 df-mre 16849 df-mri 16851 |
This theorem is referenced by: mrieqv2d 16902 mreexmrid 16906 mreexexlem2d 16908 acsfiindd 17779 |
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