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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 17633 | . 2 ⊢ (𝐶 ∈ (Moore‘𝑋) ↔ (𝐶 ⊆ 𝒫 𝑋 ∧ 𝑋 ∈ 𝐶 ∧ ∀𝑠 ∈ 𝒫 𝐶(𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝐶))) | |
| 2 | 1 | simp2bi 1147 | 1 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝑋 ∈ 𝐶) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 ≠ wne 2940 ∀wral 3061 ⊆ wss 3951 ∅c0 4333 𝒫 cpw 4600 ∩ cint 4946 ‘cfv 6561 Moorecmre 17625 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-mre 17629 |
| This theorem is referenced by: mrerintcl 17640 mreriincl 17641 mreuni 17643 mremre 17647 mrcflem 17649 mrcval 17653 mrccl 17654 mrcun 17665 mrelatglb0 18606 mreclatBAD 18608 mretopd 23100 mreclat 48886 |
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