Theorem ssuni 4875

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 4855	. . . . . 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 1918	. . 3 ⊢ (𝐵 ∈ 𝐶 → (∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵) → ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ ∪ 𝐶)))
5		df-ss 3906	. . 3 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
6		df-ss 3906	. . 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 1540   ∈ wcel 2114   ⊆ wss 3889  ∪ cuni 4850

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-v 3431  df-ss 3906  df-uni 4851

This theorem is referenced by:  elssuni  4881  uniss2  4884  ssorduni  7733  filssufilg  23876  alexsubALTlem2  24013  utoptop  24199  locfinreflem  33984  bj-sselpwuni  37357  setrec1  50166