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Theorem ismred 17822

Description: Properties that determine a Moore collection. (Contributed by Stefan O'Rear, 30-Jan-2015.)

**Hypotheses**
Ref	Expression
ismred.ss	⊢ (𝜑 → 𝐶 ⊆ 𝒫 𝑋)
ismred.ba	⊢ (𝜑 → 𝑋 ∈ 𝐶)
ismred.in	⊢ ((𝜑 ∧ 𝑠 ⊆ 𝐶 ∧ 𝑠 ≠ ∅) → ∩ 𝑠 ∈ 𝐶)

**Assertion**
Ref	Expression
ismred	⊢ (𝜑 → 𝐶 ∈ (Moore‘𝑋))

Distinct variable groups: 𝜑,𝑠 𝐶,𝑠 𝑋,𝑠

**Proof of Theorem ismred**
Step	Hyp	Ref	Expression
1		ismred.ss	. 2 ⊢ (𝜑 → 𝐶 ⊆ 𝒫 𝑋)
2		ismred.ba	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝐶)
3		velpw 4565	. . . 4 ⊢ (𝑠 ∈ 𝒫 𝐶 ↔ 𝑠 ⊆ 𝐶)
4		ismred.in	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ⊆ 𝐶 ∧ 𝑠 ≠ ∅) → ∩ 𝑠 ∈ 𝐶)
5	4	3expia 1139	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ⊆ 𝐶) → (𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝐶))
6	3, 5	sylan2b 606	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐶) → (𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝐶))
7	6	ralrimiva 3158	. 2 ⊢ (𝜑 → ∀𝑠 ∈ 𝒫 𝐶(𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝐶))
8		ismre 17810	. 2 ⊢ (𝐶 ∈ (Moore‘𝑋) ↔ (𝐶 ⊆ 𝒫 𝑋 ∧ 𝑋 ∈ 𝐶 ∧ ∀𝑠 ∈ 𝒫 𝐶(𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝐶)))
9	1, 2, 7, 8	syl3anbrc 1362	1 ⊢ (𝜑 → 𝐶 ∈ (Moore‘𝑋))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ w3a 1103 ∈ wcel 2147 ≠ wne 2959 ∀wral 3080 ⊆ wss 3900 ∅c0 4281 𝒫 cpw 4560 ∩ cint 4912 ‘cfv 6566 Moorecmre 17802

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-sep 5255 ax-nul 5266 ax-pow 5334 ax-pr 5398

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-ral 3081 df-rex 3091 df-rab 3417 df-v 3456 df-ss 3901 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4841 df-uni 4873 df-br 5109 df-opab 5174 df-mpt 5193 df-id 5553 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-iota 6520 df-fun 6568 df-fv 6574 df-mre 17806

This theorem is used by: ismred2 17823 mremre 17824 submre 17825 subrngmre 20865 subrgmre 20900 lssmre 21292 cssmre 22054 cldmre 23451 toponmre 23466 zartopn 34618 ismrcd1 43975

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