Theorem ssuni 4899

Description: Subclass relationship for class union. (Contributed by NM, 24-May-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) (Proof shortened by JJ, 26-Jul-2021.)

Assertion

Ref	Expression
ssuni	⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ⊆ ∪ 𝐶)

Proof of Theorem ssuni

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elunii 4879	. . . . . 6 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝑥 ∈ ∪ 𝐶)
2	1	expcom 413	. . . . 5 ⊢ (𝐵 ∈ 𝐶 → (𝑥 ∈ 𝐵 → 𝑥 ∈ ∪ 𝐶))
3	2	imim2d 57	. . . 4 ⊢ (𝐵 ∈ 𝐶 → ((𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵) → (𝑥 ∈ 𝐴 → 𝑥 ∈ ∪ 𝐶)))
4	3	alimdv 1916	. . 3 ⊢ (𝐵 ∈ 𝐶 → (∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵) → ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ ∪ 𝐶)))
5		df-ss 3934	. . 3 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
6		df-ss 3934	. . 3 ⊢ (𝐴 ⊆ ∪ 𝐶 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ ∪ 𝐶))
7	4, 5, 6	3imtr4g 296	. 2 ⊢ (𝐵 ∈ 𝐶 → (𝐴 ⊆ 𝐵 → 𝐴 ⊆ ∪ 𝐶))
8	7	impcom 407	1 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ⊆ ∪ 𝐶)

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1538   ∈ wcel 2109   ⊆ wss 3917  ∪ cuni 4874

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-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-v 3452  df-ss 3934  df-uni 4875

This theorem is referenced by:  elssuni  4904  uniss2  4908  ssorduni  7758  filssufilg  23805  alexsubALTlem2  23942  utoptop  24129  locfinreflem  33837  bj-sselpwuni  37045  setrec1  49684