Theorem difunieq 36243

Description: The difference of unions is a subset of the union of the difference. (Contributed by ML, 29-Mar-2021.)

Assertion

Ref	Expression
difunieq	⊢ (∪ 𝐴 ∖ ∪ 𝐵) ⊆ ∪ (𝐴 ∖ 𝐵)

Proof of Theorem difunieq

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eluni 4910	. . . 4 ⊢ (𝑥 ∈ ∪ 𝐴 ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴))
2		eluni 4910	. . . . 5 ⊢ (𝑥 ∈ ∪ 𝐵 ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵))
3	2	notbii 319	. . . 4 ⊢ (¬ 𝑥 ∈ ∪ 𝐵 ↔ ¬ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵))
4		alinexa 1845	. . . . . 6 ⊢ (∀𝑦(𝑥 ∈ 𝑦 → ¬ 𝑦 ∈ 𝐵) ↔ ¬ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵))
5		nfa1 2148	. . . . . . 7 ⊢ Ⅎ𝑦∀𝑦(𝑥 ∈ 𝑦 → ¬ 𝑦 ∈ 𝐵)
6		sp 2176	. . . . . . . . . 10 ⊢ (∀𝑦(𝑥 ∈ 𝑦 → ¬ 𝑦 ∈ 𝐵) → (𝑥 ∈ 𝑦 → ¬ 𝑦 ∈ 𝐵))
7	6	adantrd 492	. . . . . . . . 9 ⊢ (∀𝑦(𝑥 ∈ 𝑦 → ¬ 𝑦 ∈ 𝐵) → ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → ¬ 𝑦 ∈ 𝐵))
8	7	ancld 551	. . . . . . . 8 ⊢ (∀𝑦(𝑥 ∈ 𝑦 → ¬ 𝑦 ∈ 𝐵) → ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) ∧ ¬ 𝑦 ∈ 𝐵)))
9		anass 469	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) ∧ ¬ 𝑦 ∈ 𝐵) ↔ (𝑥 ∈ 𝑦 ∧ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝐵)))
10	8, 9	imbitrdi 250	. . . . . . 7 ⊢ (∀𝑦(𝑥 ∈ 𝑦 → ¬ 𝑦 ∈ 𝐵) → ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → (𝑥 ∈ 𝑦 ∧ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝐵))))
11	5, 10	eximd 2209	. . . . . 6 ⊢ (∀𝑦(𝑥 ∈ 𝑦 → ¬ 𝑦 ∈ 𝐵) → (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → ∃𝑦(𝑥 ∈ 𝑦 ∧ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝐵))))
12	4, 11	sylbir 234	. . . . 5 ⊢ (¬ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵) → (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → ∃𝑦(𝑥 ∈ 𝑦 ∧ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝐵))))
13	12	impcom 408	. . . 4 ⊢ ((∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) ∧ ¬ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)) → ∃𝑦(𝑥 ∈ 𝑦 ∧ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝐵)))
14	1, 3, 13	syl2anb 598	. . 3 ⊢ ((𝑥 ∈ ∪ 𝐴 ∧ ¬ 𝑥 ∈ ∪ 𝐵) → ∃𝑦(𝑥 ∈ 𝑦 ∧ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝐵)))
15		eldif 3957	. . 3 ⊢ (𝑥 ∈ (∪ 𝐴 ∖ ∪ 𝐵) ↔ (𝑥 ∈ ∪ 𝐴 ∧ ¬ 𝑥 ∈ ∪ 𝐵))
16		eluni 4910	. . . 4 ⊢ (𝑥 ∈ ∪ (𝐴 ∖ 𝐵) ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ (𝐴 ∖ 𝐵)))
17		eldif 3957	. . . . . 6 ⊢ (𝑦 ∈ (𝐴 ∖ 𝐵) ↔ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝐵))
18	17	anbi2i 623	. . . . 5 ⊢ ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ (𝐴 ∖ 𝐵)) ↔ (𝑥 ∈ 𝑦 ∧ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝐵)))
19	18	exbii 1850	. . . 4 ⊢ (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ (𝐴 ∖ 𝐵)) ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝐵)))
20	16, 19	bitri 274	. . 3 ⊢ (𝑥 ∈ ∪ (𝐴 ∖ 𝐵) ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝐵)))
21	14, 15, 20	3imtr4i 291	. 2 ⊢ (𝑥 ∈ (∪ 𝐴 ∖ ∪ 𝐵) → 𝑥 ∈ ∪ (𝐴 ∖ 𝐵))
22	21	ssriv 3985	1 ⊢ (∪ 𝐴 ∖ ∪ 𝐵) ⊆ ∪ (𝐴 ∖ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396  ∀wal 1539  ∃wex 1781   ∈ wcel 2106   ∖ cdif 3944   ⊆ wss 3947  ∪ cuni 4907

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-10 2137  ax-12 2171  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-v 3476  df-dif 3950  df-in 3954  df-ss 3964  df-uni 4908

This theorem is referenced by:  inunissunidif  36244