Theorem undif3ss 3332

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 3212	. . . 4 ⊢ (𝑥 ∈ (𝐴 ∪ (𝐵 ∖ 𝐶)) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ (𝐵 ∖ 𝐶)))
2		eldif 3075	. . . . 5 ⊢ (𝑥 ∈ (𝐵 ∖ 𝐶) ↔ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶))
3	2	orbi2i 751	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ (𝐵 ∖ 𝐶)) ↔ (𝑥 ∈ 𝐴 ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)))
4		orc 701	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵))
5		olc 700	. . . . . . 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 700	. . . . . . 7 ⊢ (𝑥 ∈ 𝐵 → (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵))
8		orc 701	. . . . . . 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 705	. . . . 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 722	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐶) → (𝑥 ∈ 𝐴 ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)))
13		olc 700	. . . . . 6 ⊢ ((𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶) → (𝑥 ∈ 𝐴 ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)))
14		orc 701	. . . . . . 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 948	. . . . 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 3212	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵))
21	20	biimpri 132	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) → 𝑥 ∈ (𝐴 ∪ 𝐵))
22		pm4.53r 740	. . . . . 6 ⊢ ((¬ 𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐴) → ¬ (𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐴))
23		eldif 3075	. . . . . 6 ⊢ (𝑥 ∈ (𝐶 ∖ 𝐴) ↔ (𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐴))
24	22, 23	sylnibr 666	. . . . 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 3075	. . . 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 3096	1 ⊢ (𝐴 ∪ (𝐵 ∖ 𝐶)) ⊆ ((𝐴 ∪ 𝐵) ∖ (𝐶 ∖ 𝐴))

Syntax hints:  ¬ wn 3   ∧ wa 103   ∨ wo 697   ∈ wcel 1480   ∖ cdif 3063   ∪ cun 3064   ⊆ wss 3066

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079

This theorem is referenced by: (None)