Theorem ssuni 4896

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 419	. . . . 5 ⊢ (𝐵 ∈ 𝐶 → (𝑥 ∈ 𝐵 → 𝑥 ∈ ∪ 𝐶))
3	2	imim2d 58	. . . 4 ⊢ (𝐵 ∈ 𝐶 → ((𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵) → (𝑥 ∈ 𝐴 → 𝑥 ∈ ∪ 𝐶)))
4	3	alimdv 1951	. . 3 ⊢ (𝐵 ∈ 𝐶 → (∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵) → ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ ∪ 𝐶)))
5		df-ss 3901	. . 3 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
6		df-ss 3901	. . 3 ⊢ (𝐴 ⊆ ∪ 𝐶 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ ∪ 𝐶))
7	4, 5, 6	3imtr4g 299	. 2 ⊢ (𝐵 ∈ 𝐶 → (𝐴 ⊆ 𝐵 → 𝐴 ⊆ ∪ 𝐶))
8	7	impcom 413	1 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ⊆ ∪ 𝐶)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∀wal 1568   ∈ wcel 2147   ⊆ wss 3900  ∪ cuni 4870

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-v 3456  df-ss 3901  df-uni 4871

This theorem is used by:  elssuni  4902  uniss2  4905  ssorduni  7817  setrec1  9999  filssufilg  24260  alexsubALTlem2  24397  utoptop  24583  locfinreflem  34542  bj-sselpwuni  38156