Theorem difunieq 34658

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 4841	. . . 4 ⊢ (𝑥 ∈ ∪ 𝐴 ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴))
2		eluni 4841	. . . . 5 ⊢ (𝑥 ∈ ∪ 𝐵 ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵))
3	2	notbii 322	. . . 4 ⊢ (¬ 𝑥 ∈ ∪ 𝐵 ↔ ¬ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵))
4		alinexa 1843	. . . . . 6 ⊢ (∀𝑦(𝑥 ∈ 𝑦 → ¬ 𝑦 ∈ 𝐵) ↔ ¬ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵))
5		nfa1 2155	. . . . . . 7 ⊢ Ⅎ𝑦∀𝑦(𝑥 ∈ 𝑦 → ¬ 𝑦 ∈ 𝐵)
6		sp 2182	. . . . . . . . . 10 ⊢ (∀𝑦(𝑥 ∈ 𝑦 → ¬ 𝑦 ∈ 𝐵) → (𝑥 ∈ 𝑦 → ¬ 𝑦 ∈ 𝐵))
7	6	adantrd 494	. . . . . . . . 9 ⊢ (∀𝑦(𝑥 ∈ 𝑦 → ¬ 𝑦 ∈ 𝐵) → ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → ¬ 𝑦 ∈ 𝐵))
8	7	ancld 553	. . . . . . . 8 ⊢ (∀𝑦(𝑥 ∈ 𝑦 → ¬ 𝑦 ∈ 𝐵) → ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) ∧ ¬ 𝑦 ∈ 𝐵)))
9		anass 471	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) ∧ ¬ 𝑦 ∈ 𝐵) ↔ (𝑥 ∈ 𝑦 ∧ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝐵)))
10	8, 9	syl6ib 253	. . . . . . 7 ⊢ (∀𝑦(𝑥 ∈ 𝑦 → ¬ 𝑦 ∈ 𝐵) → ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → (𝑥 ∈ 𝑦 ∧ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝐵))))
11	5, 10	eximd 2216	. . . . . 6 ⊢ (∀𝑦(𝑥 ∈ 𝑦 → ¬ 𝑦 ∈ 𝐵) → (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → ∃𝑦(𝑥 ∈ 𝑦 ∧ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝐵))))
12	4, 11	sylbir 237	. . . . 5 ⊢ (¬ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵) → (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → ∃𝑦(𝑥 ∈ 𝑦 ∧ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝐵))))
13	12	impcom 410	. . . 4 ⊢ ((∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) ∧ ¬ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)) → ∃𝑦(𝑥 ∈ 𝑦 ∧ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝐵)))
14	1, 3, 13	syl2anb 599	. . 3 ⊢ ((𝑥 ∈ ∪ 𝐴 ∧ ¬ 𝑥 ∈ ∪ 𝐵) → ∃𝑦(𝑥 ∈ 𝑦 ∧ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝐵)))
15		eldif 3946	. . 3 ⊢ (𝑥 ∈ (∪ 𝐴 ∖ ∪ 𝐵) ↔ (𝑥 ∈ ∪ 𝐴 ∧ ¬ 𝑥 ∈ ∪ 𝐵))
16		eluni 4841	. . . 4 ⊢ (𝑥 ∈ ∪ (𝐴 ∖ 𝐵) ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ (𝐴 ∖ 𝐵)))
17		eldif 3946	. . . . . 6 ⊢ (𝑦 ∈ (𝐴 ∖ 𝐵) ↔ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝐵))
18	17	anbi2i 624	. . . . 5 ⊢ ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ (𝐴 ∖ 𝐵)) ↔ (𝑥 ∈ 𝑦 ∧ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝐵)))
19	18	exbii 1848	. . . 4 ⊢ (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ (𝐴 ∖ 𝐵)) ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝐵)))
20	16, 19	bitri 277	. . 3 ⊢ (𝑥 ∈ ∪ (𝐴 ∖ 𝐵) ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝐵)))
21	14, 15, 20	3imtr4i 294	. 2 ⊢ (𝑥 ∈ (∪ 𝐴 ∖ ∪ 𝐵) → 𝑥 ∈ ∪ (𝐴 ∖ 𝐵))
22	21	ssriv 3971	1 ⊢ (∪ 𝐴 ∖ ∪ 𝐵) ⊆ ∪ (𝐴 ∖ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398  ∀wal 1535  ∃wex 1780   ∈ wcel 2114   ∖ cdif 3933   ⊆ wss 3936  ∪ cuni 4838

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-v 3496  df-dif 3939  df-in 3943  df-ss 3952  df-uni 4839

This theorem is referenced by:  inunissunidif  34659