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Mirrors > Home > MPE Home > Th. List > mriss | Structured version Visualization version GIF version |
Description: An independent set of a Moore system is a subset of the base set. (Contributed by David Moews, 1-May-2017.) |
Ref | Expression |
---|---|
mriss.1 | ⊢ 𝐼 = (mrInd‘𝐴) |
Ref | Expression |
---|---|
mriss | ⊢ ((𝐴 ∈ (Moore‘𝑋) ∧ 𝑆 ∈ 𝐼) → 𝑆 ⊆ 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2826 | . . 3 ⊢ (mrCls‘𝐴) = (mrCls‘𝐴) | |
2 | mriss.1 | . . 3 ⊢ 𝐼 = (mrInd‘𝐴) | |
3 | 1, 2 | ismri 16645 | . 2 ⊢ (𝐴 ∈ (Moore‘𝑋) → (𝑆 ∈ 𝐼 ↔ (𝑆 ⊆ 𝑋 ∧ ∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ ((mrCls‘𝐴)‘(𝑆 ∖ {𝑥}))))) |
4 | 3 | simprbda 494 | 1 ⊢ ((𝐴 ∈ (Moore‘𝑋) ∧ 𝑆 ∈ 𝐼) → 𝑆 ⊆ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 386 = wceq 1658 ∈ wcel 2166 ∀wral 3118 ∖ cdif 3796 ⊆ wss 3799 {csn 4398 ‘cfv 6124 Moorecmre 16596 mrClscmrc 16597 mrIndcmri 16598 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-ral 3123 df-rex 3124 df-rab 3127 df-v 3417 df-sbc 3664 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4660 df-br 4875 df-opab 4937 df-mpt 4954 df-id 5251 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-iota 6087 df-fun 6126 df-fv 6132 df-mre 16600 df-mri 16602 |
This theorem is referenced by: mrissd 16650 |
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