Theorem undif3ss 3388

Description: A subset relationship involving class union and class difference. In classical logic, this would be equality rather than subset, as in the first equality of Exercise 13 of [TakeutiZaring] p. 22. (Contributed by Jim Kingdon, 28-Jul-2018.)

Assertion

Ref	Expression
undif3ss	⊢ (𝐴 ∪ (𝐵 ∖ 𝐶)) ⊆ ((𝐴 ∪ 𝐵) ∖ (𝐶 ∖ 𝐴))

Proof of Theorem undif3ss

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elun 3268	. . . 4 ⊢ (𝑥 ∈ (𝐴 ∪ (𝐵 ∖ 𝐶)) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ (𝐵 ∖ 𝐶)))
2		eldif 3130	. . . . 5 ⊢ (𝑥 ∈ (𝐵 ∖ 𝐶) ↔ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶))
3	2	orbi2i 757	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ (𝐵 ∖ 𝐶)) ↔ (𝑥 ∈ 𝐴 ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)))
4		orc 707	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵))
5		olc 706	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → (¬ 𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐴))
6	4, 5	jca 304	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 → ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ∧ (¬ 𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐴)))
7		olc 706	. . . . . . 7 ⊢ (𝑥 ∈ 𝐵 → (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵))
8		orc 707	. . . . . . 7 ⊢ (¬ 𝑥 ∈ 𝐶 → (¬ 𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐴))
9	7, 8	anim12i 336	. . . . . 6 ⊢ ((𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶) → ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ∧ (¬ 𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐴)))
10	6, 9	jaoi 711	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)) → ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ∧ (¬ 𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐴)))
11		simpl 108	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐶) → 𝑥 ∈ 𝐴)
12	11	orcd 728	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐶) → (𝑥 ∈ 𝐴 ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)))
13		olc 706	. . . . . 6 ⊢ ((𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶) → (𝑥 ∈ 𝐴 ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)))
14		orc 707	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐴 ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)))
15	14	adantr 274	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ 𝐴 ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)))
16	14	adantl 275	. . . . . 6 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ 𝐴 ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)))
17	12, 13, 15, 16	ccase 959	. . . . 5 ⊢ (((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ∧ (¬ 𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐴)) → (𝑥 ∈ 𝐴 ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)))
18	10, 17	impbii 125	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)) ↔ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ∧ (¬ 𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐴)))
19	1, 3, 18	3bitri 205	. . 3 ⊢ (𝑥 ∈ (𝐴 ∪ (𝐵 ∖ 𝐶)) ↔ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ∧ (¬ 𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐴)))
20		elun 3268	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵))
21	20	biimpri 132	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) → 𝑥 ∈ (𝐴 ∪ 𝐵))
22		pm4.53r 746	. . . . . 6 ⊢ ((¬ 𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐴) → ¬ (𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐴))
23		eldif 3130	. . . . . 6 ⊢ (𝑥 ∈ (𝐶 ∖ 𝐴) ↔ (𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐴))
24	22, 23	sylnibr 672	. . . . 5 ⊢ ((¬ 𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐴) → ¬ 𝑥 ∈ (𝐶 ∖ 𝐴))
25	21, 24	anim12i 336	. . . 4 ⊢ (((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ∧ (¬ 𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐴)) → (𝑥 ∈ (𝐴 ∪ 𝐵) ∧ ¬ 𝑥 ∈ (𝐶 ∖ 𝐴)))
26		eldif 3130	. . . 4 ⊢ (𝑥 ∈ ((𝐴 ∪ 𝐵) ∖ (𝐶 ∖ 𝐴)) ↔ (𝑥 ∈ (𝐴 ∪ 𝐵) ∧ ¬ 𝑥 ∈ (𝐶 ∖ 𝐴)))
27	25, 26	sylibr 133	. . 3 ⊢ (((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ∧ (¬ 𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐴)) → 𝑥 ∈ ((𝐴 ∪ 𝐵) ∖ (𝐶 ∖ 𝐴)))
28	19, 27	sylbi 120	. 2 ⊢ (𝑥 ∈ (𝐴 ∪ (𝐵 ∖ 𝐶)) → 𝑥 ∈ ((𝐴 ∪ 𝐵) ∖ (𝐶 ∖ 𝐴)))
29	28	ssriv 3151	1 ⊢ (𝐴 ∪ (𝐵 ∖ 𝐶)) ⊆ ((𝐴 ∪ 𝐵) ∖ (𝐶 ∖ 𝐴))

Syntax hints:  ¬ wn 3   ∧ wa 103   ∨ wo 703   ∈ wcel 2141   ∖ cdif 3118   ∪ cun 3119   ⊆ wss 3121

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134

This theorem is referenced by: (None)