Theorem ssuni 4872

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 4849	. . . . . 6 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝑥 ∈ ∪ 𝐶)
2	1	expcom 415	. . . . 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 1917	. . 3 ⊢ (𝐵 ∈ 𝐶 → (∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵) → ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ ∪ 𝐶)))
5		dfss2 3912	. . 3 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
6		dfss2 3912	. . 3 ⊢ (𝐴 ⊆ ∪ 𝐶 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ ∪ 𝐶))
7	4, 5, 6	3imtr4g 296	. 2 ⊢ (𝐵 ∈ 𝐶 → (𝐴 ⊆ 𝐵 → 𝐴 ⊆ ∪ 𝐶))
8	7	impcom 409	1 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ⊆ ∪ 𝐶)

Syntax hints:   → wi 4   ∧ wa 397  ∀wal 1537   ∈ wcel 2104   ⊆ wss 3892  ∪ cuni 4844

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2707

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1542  df-ex 1780  df-sb 2066  df-clab 2714  df-cleq 2728  df-clel 2814  df-v 3439  df-in 3899  df-ss 3909  df-uni 4845

This theorem is referenced by:  elssuni  4877  uniss2  4881  ssorduni  7661  filssufilg  23111  alexsubALTlem2  23248  utoptop  23435  locfinreflem  31839  bj-sselpwuni  35271  setrec1  46641