Theorem difunieq 38217

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 4869	. . . 4 ⊢ (𝑥 ∈ ∪ 𝐴 ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴))
2		eluni 4869	. . . . 5 ⊢ (𝑥 ∈ ∪ 𝐵 ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵))
3	2	notbii 323	. . . 4 ⊢ (¬ 𝑥 ∈ ∪ 𝐵 ↔ ¬ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵))
4		alinexa 1876	. . . . . 6 ⊢ (∀𝑦(𝑥 ∈ 𝑦 → ¬ 𝑦 ∈ 𝐵) ↔ ¬ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵))
5		nfa1 2188	. . . . . . 7 ⊢ Ⅎ𝑦∀𝑦(𝑥 ∈ 𝑦 → ¬ 𝑦 ∈ 𝐵)
6		sp 2219	. . . . . . . . . 10 ⊢ (∀𝑦(𝑥 ∈ 𝑦 → ¬ 𝑦 ∈ 𝐵) → (𝑥 ∈ 𝑦 → ¬ 𝑦 ∈ 𝐵))
7	6	adantrd 497	. . . . . . . . 9 ⊢ (∀𝑦(𝑥 ∈ 𝑦 → ¬ 𝑦 ∈ 𝐵) → ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → ¬ 𝑦 ∈ 𝐵))
8	7	ancld 560	. . . . . . . 8 ⊢ (∀𝑦(𝑥 ∈ 𝑦 → ¬ 𝑦 ∈ 𝐵) → ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) ∧ ¬ 𝑦 ∈ 𝐵)))
9		anass 474	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) ∧ ¬ 𝑦 ∈ 𝐵) ↔ (𝑥 ∈ 𝑦 ∧ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝐵)))
10	8, 9	imbitrdi 254	. . . . . . 7 ⊢ (∀𝑦(𝑥 ∈ 𝑦 → ¬ 𝑦 ∈ 𝐵) → ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → (𝑥 ∈ 𝑦 ∧ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝐵))))
11	5, 10	eximd 2252	. . . . . 6 ⊢ (∀𝑦(𝑥 ∈ 𝑦 → ¬ 𝑦 ∈ 𝐵) → (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → ∃𝑦(𝑥 ∈ 𝑦 ∧ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝐵))))
12	4, 11	sylbir 238	. . . . 5 ⊢ (¬ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵) → (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → ∃𝑦(𝑥 ∈ 𝑦 ∧ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝐵))))
13	12	impcom 413	. . . 4 ⊢ ((∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) ∧ ¬ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)) → ∃𝑦(𝑥 ∈ 𝑦 ∧ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝐵)))
14	1, 3, 13	syl2anb 610	. . 3 ⊢ ((𝑥 ∈ ∪ 𝐴 ∧ ¬ 𝑥 ∈ ∪ 𝐵) → ∃𝑦(𝑥 ∈ 𝑦 ∧ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝐵)))
15		eldif 3908	. . 3 ⊢ (𝑥 ∈ (∪ 𝐴 ∖ ∪ 𝐵) ↔ (𝑥 ∈ ∪ 𝐴 ∧ ¬ 𝑥 ∈ ∪ 𝐵))
16		eluni 4869	. . . 4 ⊢ (𝑥 ∈ ∪ (𝐴 ∖ 𝐵) ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ (𝐴 ∖ 𝐵)))
17		eldif 3908	. . . . . 6 ⊢ (𝑦 ∈ (𝐴 ∖ 𝐵) ↔ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝐵))
18	17	anbi2i 635	. . . . 5 ⊢ ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ (𝐴 ∖ 𝐵)) ↔ (𝑥 ∈ 𝑦 ∧ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝐵)))
19	18	exbii 1881	. . . 4 ⊢ (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ (𝐴 ∖ 𝐵)) ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝐵)))
20	16, 19	bitri 278	. . 3 ⊢ (𝑥 ∈ ∪ (𝐴 ∖ 𝐵) ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝐵)))
21	14, 15, 20	3imtr4i 295	. 2 ⊢ (𝑥 ∈ (∪ 𝐴 ∖ ∪ 𝐵) → 𝑥 ∈ ∪ (𝐴 ∖ 𝐵))
22	21	ssriv 3934	1 ⊢ (∪ 𝐴 ∖ ∪ 𝐵) ⊆ ∪ (𝐴 ∖ 𝐵)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401  ∀wal 1568  ∃wex 1812   ∈ wcel 2145   ∖ cdif 3895   ⊆ wss 3898  ∪ cuni 4866

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-12 2213  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-dif 3901  df-ss 3915  df-uni 4867

This theorem is used by:  inunissunidif  38218