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Mirrors > Home > MPE Home > Th. List > mress | Structured version Visualization version GIF version |
Description: A Moore-closed subset is a subset. (Contributed by Stefan O'Rear, 31-Jan-2015.) |
Ref | Expression |
---|---|
mress | ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ∈ 𝐶) → 𝑆 ⊆ 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mresspw 16969 | . . 3 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐶 ⊆ 𝒫 𝑋) | |
2 | 1 | sselda 3878 | . 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ∈ 𝐶) → 𝑆 ∈ 𝒫 𝑋) |
3 | 2 | elpwid 4500 | 1 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ∈ 𝐶) → 𝑆 ⊆ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∈ wcel 2114 ⊆ wss 3844 𝒫 cpw 4489 ‘cfv 6340 Moorecmre 16959 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-sep 5168 ax-nul 5175 ax-pow 5233 ax-pr 5297 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3401 df-sbc 3682 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-nul 4213 df-if 4416 df-pw 4491 df-sn 4518 df-pr 4520 df-op 4524 df-uni 4798 df-br 5032 df-opab 5094 df-mpt 5112 df-id 5430 df-xp 5532 df-rel 5533 df-cnv 5534 df-co 5535 df-dm 5536 df-iota 6298 df-fun 6342 df-fv 6348 df-mre 16963 |
This theorem is referenced by: mreriincl 16975 mrcid 16990 mrcsscl 16997 submrc 17005 acsfiel 17031 mrelatglb0 17914 isnacs3 40127 |
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