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Mirrors > Home > MPE Home > Th. List > mre1cl | Structured version Visualization version GIF version |
Description: In any Moore collection the base set is closed. (Contributed by Stefan O'Rear, 30-Jan-2015.) |
Ref | Expression |
---|---|
mre1cl | ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝑋 ∈ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ismre 16610 | . 2 ⊢ (𝐶 ∈ (Moore‘𝑋) ↔ (𝐶 ⊆ 𝒫 𝑋 ∧ 𝑋 ∈ 𝐶 ∧ ∀𝑠 ∈ 𝒫 𝐶(𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝐶))) | |
2 | 1 | simp2bi 1180 | 1 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝑋 ∈ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2164 ≠ wne 2999 ∀wral 3117 ⊆ wss 3798 ∅c0 4146 𝒫 cpw 4380 ∩ cint 4699 ‘cfv 6127 Moorecmre 16602 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-br 4876 df-opab 4938 df-mpt 4955 df-id 5252 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-iota 6090 df-fun 6129 df-fv 6135 df-mre 16606 |
This theorem is referenced by: mrerintcl 16617 mreriincl 16618 mreuni 16620 mremre 16624 mrcflem 16626 mrcval 16630 mrccl 16631 mrcun 16642 mrelatglb0 17545 mreclatBAD 17547 mretopd 21274 |
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