Theorem ssuni 4612

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 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elunii 4594	. . . . . 6 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝑦 ∈ ∪ 𝐶)
2	1	expcom 450	. . . . 5 ⊢ (𝐵 ∈ 𝐶 → (𝑦 ∈ 𝐵 → 𝑦 ∈ ∪ 𝐶))
3	2	imim2d 57	. . . 4 ⊢ (𝐵 ∈ 𝐶 → ((𝑦 ∈ 𝐴 → 𝑦 ∈ 𝐵) → (𝑦 ∈ 𝐴 → 𝑦 ∈ ∪ 𝐶)))
4	3	alimdv 1995	. . 3 ⊢ (𝐵 ∈ 𝐶 → (∀𝑦(𝑦 ∈ 𝐴 → 𝑦 ∈ 𝐵) → ∀𝑦(𝑦 ∈ 𝐴 → 𝑦 ∈ ∪ 𝐶)))
5		dfss2 3733	. . 3 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑦(𝑦 ∈ 𝐴 → 𝑦 ∈ 𝐵))
6		dfss2 3733	. . 3 ⊢ (𝐴 ⊆ ∪ 𝐶 ↔ ∀𝑦(𝑦 ∈ 𝐴 → 𝑦 ∈ ∪ 𝐶))
7	4, 5, 6	3imtr4g 285	. 2 ⊢ (𝐵 ∈ 𝐶 → (𝐴 ⊆ 𝐵 → 𝐴 ⊆ ∪ 𝐶))
8	7	impcom 445	1 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ⊆ ∪ 𝐶)

Syntax hints:   → wi 4   ∧ wa 383  ∀wal 1630   ∈ wcel 2140   ⊆ wss 3716  ∪ cuni 4589

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-v 3343  df-in 3723  df-ss 3730  df-uni 4590

This theorem is referenced by:  elssuni  4620  uniss2  4623  ssorduni  7152  filssufilg  21937  alexsubALTlem2  22074  utoptop  22260  locfinreflem  30238  setrec1  42967