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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 17635 | . 2 ⊢ (𝐶 ∈ (Moore‘𝑋) ↔ (𝐶 ⊆ 𝒫 𝑋 ∧ 𝑋 ∈ 𝐶 ∧ ∀𝑠 ∈ 𝒫 𝐶(𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝐶))) | |
2 | 1 | simp2bi 1145 | 1 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝑋 ∈ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 ≠ wne 2938 ∀wral 3059 ⊆ wss 3963 ∅c0 4339 𝒫 cpw 4605 ∩ cint 4951 ‘cfv 6563 Moorecmre 17627 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-iota 6516 df-fun 6565 df-fv 6571 df-mre 17631 |
This theorem is referenced by: mrerintcl 17642 mreriincl 17643 mreuni 17645 mremre 17649 mrcflem 17651 mrcval 17655 mrccl 17656 mrcun 17667 mrelatglb0 18619 mreclatBAD 18621 mretopd 23116 mreclat 48786 |
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