Theorem psdmullem 22519

Description: Lemma for psdmul 22520. Transitive law for union of class difference. (Contributed by SN, 5-May-2025.)

Hypotheses

Ref	Expression
psdmullem.cb	⊢ (𝜑 → 𝐶 ⊆ 𝐵)
psdmullem.ba	⊢ (𝜑 → 𝐵 ⊆ 𝐴)

Assertion

Ref	Expression
psdmullem	⊢ (𝜑 → ((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐶)) = (𝐴 ∖ 𝐶))

Proof of Theorem psdmullem

Step	Hyp	Ref	Expression
1		undif3 4248	. 2 ⊢ ((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐶)) = (((𝐴 ∖ 𝐵) ∪ 𝐵) ∖ (𝐶 ∖ (𝐴 ∖ 𝐵)))
2		psdmullem.ba	. . . 4 ⊢ (𝜑 → 𝐵 ⊆ 𝐴)
3		undifr 4442	. . . 4 ⊢ (𝐵 ⊆ 𝐴 ↔ ((𝐴 ∖ 𝐵) ∪ 𝐵) = 𝐴)
4	2, 3	sylib 221	. . 3 ⊢ (𝜑 → ((𝐴 ∖ 𝐵) ∪ 𝐵) = 𝐴)
5		difdif2 4244	. . . 4 ⊢ (𝐶 ∖ (𝐴 ∖ 𝐵)) = ((𝐶 ∖ 𝐴) ∪ (𝐶 ∩ 𝐵))
6		psdmullem.cb	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ⊆ 𝐵)
7	6, 2	sstrd 3923	. . . . . . 7 ⊢ (𝜑 → 𝐶 ⊆ 𝐴)
8		ssdif0 4317	. . . . . . 7 ⊢ (𝐶 ⊆ 𝐴 ↔ (𝐶 ∖ 𝐴) = ∅)
9	7, 8	sylib 221	. . . . . 6 ⊢ (𝜑 → (𝐶 ∖ 𝐴) = ∅)
10		dfss2 4097	. . . . . . 7 ⊢ (𝐶 ⊆ 𝐵 ↔ (𝐶 ∩ 𝐵) = 𝐶)
11	6, 10	sylib 221	. . . . . 6 ⊢ (𝜑 → (𝐶 ∩ 𝐵) = 𝐶)
12	9, 11	uneq12d 4171	. . . . 5 ⊢ (𝜑 → ((𝐶 ∖ 𝐴) ∪ (𝐶 ∩ 𝐵)) = (∅ ∪ 𝐶))
13		0un 4349	. . . . 5 ⊢ (∅ ∪ 𝐶) = 𝐶
14	12, 13	eqtrdi 2814	. . . 4 ⊢ (𝜑 → ((𝐶 ∖ 𝐴) ∪ (𝐶 ∩ 𝐵)) = 𝐶)
15	5, 14	eqtrid 2810	. . 3 ⊢ (𝜑 → (𝐶 ∖ (𝐴 ∖ 𝐵)) = 𝐶)
16	4, 15	difeq12d 4067	. 2 ⊢ (𝜑 → (((𝐴 ∖ 𝐵) ∪ 𝐵) ∖ (𝐶 ∖ (𝐴 ∖ 𝐵))) = (𝐴 ∖ 𝐶))
17	1, 16	eqtrid 2810	1 ⊢ (𝜑 → ((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐶)) = (𝐴 ∖ 𝐶))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ⊆ wss 3900   ∖ cdif 4048   ∩ cin 4092   ∪ cun 4152  ∅c0 4281

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282

This theorem is used by:  psdmul  22520