Theorem psdmullem 22159

Description: Lemma for psdmul 22160. 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 4292	. 2 ⊢ ((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐶)) = (((𝐴 ∖ 𝐵) ∪ 𝐵) ∖ (𝐶 ∖ (𝐴 ∖ 𝐵)))
2		psdmullem.ba	. . . 4 ⊢ (𝜑 → 𝐵 ⊆ 𝐴)
3		undifr 4487	. . . 4 ⊢ (𝐵 ⊆ 𝐴 ↔ ((𝐴 ∖ 𝐵) ∪ 𝐵) = 𝐴)
4	2, 3	sylib 217	. . 3 ⊢ (𝜑 → ((𝐴 ∖ 𝐵) ∪ 𝐵) = 𝐴)
5		difdif2 4288	. . . 4 ⊢ (𝐶 ∖ (𝐴 ∖ 𝐵)) = ((𝐶 ∖ 𝐴) ∪ (𝐶 ∩ 𝐵))
6		psdmullem.cb	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ⊆ 𝐵)
7	6, 2	sstrd 3990	. . . . . . 7 ⊢ (𝜑 → 𝐶 ⊆ 𝐴)
8		ssdif0 4366	. . . . . . 7 ⊢ (𝐶 ⊆ 𝐴 ↔ (𝐶 ∖ 𝐴) = ∅)
9	7, 8	sylib 217	. . . . . 6 ⊢ (𝜑 → (𝐶 ∖ 𝐴) = ∅)
10		dfss2 3965	. . . . . . 7 ⊢ (𝐶 ⊆ 𝐵 ↔ (𝐶 ∩ 𝐵) = 𝐶)
11	6, 10	sylib 217	. . . . . 6 ⊢ (𝜑 → (𝐶 ∩ 𝐵) = 𝐶)
12	9, 11	uneq12d 4164	. . . . 5 ⊢ (𝜑 → ((𝐶 ∖ 𝐴) ∪ (𝐶 ∩ 𝐵)) = (∅ ∪ 𝐶))
13		0un 4397	. . . . 5 ⊢ (∅ ∪ 𝐶) = 𝐶
14	12, 13	eqtrdi 2782	. . . 4 ⊢ (𝜑 → ((𝐶 ∖ 𝐴) ∪ (𝐶 ∩ 𝐵)) = 𝐶)
15	5, 14	eqtrid 2778	. . 3 ⊢ (𝜑 → (𝐶 ∖ (𝐴 ∖ 𝐵)) = 𝐶)
16	4, 15	difeq12d 4122	. 2 ⊢ (𝜑 → (((𝐴 ∖ 𝐵) ∪ 𝐵) ∖ (𝐶 ∖ (𝐴 ∖ 𝐵))) = (𝐴 ∖ 𝐶))
17	1, 16	eqtrid 2778	1 ⊢ (𝜑 → ((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐶)) = (𝐴 ∖ 𝐶))

Syntax hints:   → wi 4   = wceq 1534   ∖ cdif 3944   ∪ cun 3945   ∩ cin 3946   ⊆ wss 3947  ∅c0 4325

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326

This theorem is referenced by:  psdmul  22160