Theorem mreuniss 48816

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 4887	. . 3 ⊢ (𝑆 ⊆ 𝐶 → ∪ 𝑆 ⊆ ∪ 𝐶)
2	1	adantl 481	. 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝐶) → ∪ 𝑆 ⊆ ∪ 𝐶)
3		mreuni 17567	. . 3 ⊢ (𝐶 ∈ (Moore‘𝑋) → ∪ 𝐶 = 𝑋)
4	3	adantr 480	. 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝐶) → ∪ 𝐶 = 𝑋)
5	2, 4	sseqtrd 3991	1 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝐶) → ∪ 𝑆 ⊆ 𝑋)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ⊆ wss 3922  ∪ cuni 4879  ‘cfv 6519  Moorecmre 17549

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5259  ax-nul 5269  ax-pow 5328  ax-pr 5395

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2880  df-ral 3047  df-rex 3056  df-rab 3412  df-v 3457  df-dif 3925  df-un 3927  df-in 3929  df-ss 3939  df-nul 4305  df-if 4497  df-pw 4573  df-sn 4598  df-pr 4600  df-op 4604  df-uni 4880  df-br 5116  df-opab 5178  df-mpt 5197  df-id 5541  df-xp 5652  df-rel 5653  df-cnv 5654  df-co 5655  df-dm 5656  df-iota 6472  df-fun 6521  df-fv 6527  df-mre 17553

This theorem is referenced by:  mrelatlubALT  48911  mreclat  48913