Theorem ssuni 4898

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 4875	. . . . . 6 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝑥 ∈ ∪ 𝐶)
2	1	expcom 414	. . . . 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 1919	. . 3 ⊢ (𝐵 ∈ 𝐶 → (∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵) → ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ ∪ 𝐶)))
5		dfss2 3933	. . 3 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
6		dfss2 3933	. . 3 ⊢ (𝐴 ⊆ ∪ 𝐶 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ ∪ 𝐶))
7	4, 5, 6	3imtr4g 295	. 2 ⊢ (𝐵 ∈ 𝐶 → (𝐴 ⊆ 𝐵 → 𝐴 ⊆ ∪ 𝐶))
8	7	impcom 408	1 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ⊆ ∪ 𝐶)

Syntax hints:   → wi 4   ∧ wa 396  ∀wal 1539   ∈ wcel 2106   ⊆ wss 3913  ∪ cuni 4870

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-v 3448  df-in 3920  df-ss 3930  df-uni 4871

This theorem is referenced by:  elssuni  4903  uniss2  4907  ssorduni  7718  filssufilg  23299  alexsubALTlem2  23436  utoptop  23623  locfinreflem  32510  bj-sselpwuni  35594  setrec1  47256