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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 16907 | . . . 4 ⊢ (𝜑 → 𝑆 ⊆ 𝑋) |
6 | 2, 3, 4, 5 | ismri2d 16904 | . . 3 ⊢ (𝜑 → (𝑆 ∈ 𝐼 ↔ ∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})))) |
7 | 1, 6 | mpbid 234 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))) |
8 | ismri2dad.5 | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑆) | |
9 | simpr 487 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑌) → 𝑥 = 𝑌) | |
10 | 9 | sneqd 4579 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝑌) → {𝑥} = {𝑌}) |
11 | 10 | difeq2d 4099 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑌) → (𝑆 ∖ {𝑥}) = (𝑆 ∖ {𝑌})) |
12 | 11 | fveq2d 6674 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑌) → (𝑁‘(𝑆 ∖ {𝑥})) = (𝑁‘(𝑆 ∖ {𝑌}))) |
13 | 9, 12 | eleq12d 2907 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑌) → (𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})) ↔ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌})))) |
14 | 13 | notbid 320 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑌) → (¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})) ↔ ¬ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌})))) |
15 | 8, 14 | rspcdv 3615 | . 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 398 = wceq 1537 ∈ wcel 2114 ∀wral 3138 ∖ cdif 3933 {csn 4567 ‘cfv 6355 Moorecmre 16853 mrClscmrc 16854 mrIndcmri 16855 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-iota 6314 df-fun 6357 df-fv 6363 df-mre 16857 df-mri 16859 |
This theorem is referenced by: mrieqv2d 16910 mreexmrid 16914 mreexexlem2d 16916 acsfiindd 17787 |
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