Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 16119 | . . . 4 ⊢ (𝜑 → 𝑆 ⊆ 𝑋) |
6 | 2, 3, 4, 5 | ismri2d 16116 | . . 3 ⊢ (𝜑 → (𝑆 ∈ 𝐼 ↔ ∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})))) |
7 | 1, 6 | mpbid 221 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))) |
8 | ismri2dad.5 | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑆) | |
9 | simpr 476 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑌) → 𝑥 = 𝑌) | |
10 | 9 | sneqd 4137 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝑌) → {𝑥} = {𝑌}) |
11 | 10 | difeq2d 3690 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑌) → (𝑆 ∖ {𝑥}) = (𝑆 ∖ {𝑌})) |
12 | 11 | fveq2d 6107 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑌) → (𝑁‘(𝑆 ∖ {𝑥})) = (𝑁‘(𝑆 ∖ {𝑌}))) |
13 | 9, 12 | eleq12d 2682 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑌) → (𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})) ↔ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌})))) |
14 | 13 | notbid 307 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑌) → (¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})) ↔ ¬ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌})))) |
15 | 8, 14 | rspcdv 3285 | . 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 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ∖ cdif 3537 {csn 4125 ‘cfv 5804 Moorecmre 16065 mrClscmrc 16066 mrIndcmri 16067 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-iota 5768 df-fun 5806 df-fv 5812 df-mre 16069 df-mri 16071 |
This theorem is referenced by: mrieqv2d 16122 mreexmrid 16126 mreexexlem2d 16128 acsfiindd 17000 |
Copyright terms: Public domain | W3C validator |