Theorem mreuniss 48741

Description: The union of a collection of closed sets is a subset. (Contributed by Zhi Wang, 29-Sep-2024.)

Assertion

Ref	Expression
mreuniss	⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝐶) → ∪ 𝑆 ⊆ 𝑋)

Proof of Theorem mreuniss

Step	Hyp	Ref	Expression
1		uniss 4923	. . 3 ⊢ (𝑆 ⊆ 𝐶 → ∪ 𝑆 ⊆ ∪ 𝐶)
2	1	adantl 481	. 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝐶) → ∪ 𝑆 ⊆ ∪ 𝐶)
3		mreuni 17654	. . 3 ⊢ (𝐶 ∈ (Moore‘𝑋) → ∪ 𝐶 = 𝑋)
4	3	adantr 480	. 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝐶) → ∪ 𝐶 = 𝑋)
5	2, 4	sseqtrd 4039	1 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝐶) → ∪ 𝑆 ⊆ 𝑋)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108   ⊆ wss 3966  ∪ cuni 4915  ‘cfv 6569  Moorecmre 17636

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  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 5305  ax-nul 5315  ax-pow 5374  ax-pr 5441

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  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 3483  df-dif 3969  df-un 3971  df-in 3973  df-ss 3983  df-nul 4343  df-if 4535  df-pw 4610  df-sn 4635  df-pr 4637  df-op 4641  df-uni 4916  df-br 5152  df-opab 5214  df-mpt 5235  df-id 5587  df-xp 5699  df-rel 5700  df-cnv 5701  df-co 5702  df-dm 5703  df-iota 6522  df-fun 6571  df-fv 6577  df-mre 17640

This theorem is referenced by:  mrelatlubALT  48829  mreclat  48831