Theorem rp-fakeuninass 44104

Description: A special case where a mixture of union and intersection appears to conform to a mixed associative law. (Contributed by RP, 29-Feb-2020.)

Assertion

Ref	Expression
rp-fakeuninass	⊢ (𝐴 ⊆ 𝐶 ↔ ((𝐴 ∪ 𝐵) ∩ 𝐶) = (𝐴 ∪ (𝐵 ∩ 𝐶)))

Proof of Theorem rp-fakeuninass

Step	Hyp	Ref	Expression
1		rp-fakeinunass 44103	. 2 ⊢ (𝐴 ⊆ 𝐶 ↔ ((𝐶 ∩ 𝐵) ∪ 𝐴) = (𝐶 ∩ (𝐵 ∪ 𝐴)))
2		eqcom 2772	. 2 ⊢ (((𝐶 ∩ 𝐵) ∪ 𝐴) = (𝐶 ∩ (𝐵 ∪ 𝐴)) ↔ (𝐶 ∩ (𝐵 ∪ 𝐴)) = ((𝐶 ∩ 𝐵) ∪ 𝐴))
3		incom 4164	. . . 4 ⊢ (𝐶 ∩ (𝐵 ∪ 𝐴)) = ((𝐵 ∪ 𝐴) ∩ 𝐶)
4		uncom 4114	. . . . 5 ⊢ (𝐵 ∪ 𝐴) = (𝐴 ∪ 𝐵)
5	4	ineq1i 4171	. . . 4 ⊢ ((𝐵 ∪ 𝐴) ∩ 𝐶) = ((𝐴 ∪ 𝐵) ∩ 𝐶)
6	3, 5	eqtri 2788	. . 3 ⊢ (𝐶 ∩ (𝐵 ∪ 𝐴)) = ((𝐴 ∪ 𝐵) ∩ 𝐶)
7		uncom 4114	. . . 4 ⊢ ((𝐶 ∩ 𝐵) ∪ 𝐴) = (𝐴 ∪ (𝐶 ∩ 𝐵))
8		incom 4164	. . . . 5 ⊢ (𝐶 ∩ 𝐵) = (𝐵 ∩ 𝐶)
9	8	uneq2i 4121	. . . 4 ⊢ (𝐴 ∪ (𝐶 ∩ 𝐵)) = (𝐴 ∪ (𝐵 ∩ 𝐶))
10	7, 9	eqtri 2788	. . 3 ⊢ ((𝐶 ∩ 𝐵) ∪ 𝐴) = (𝐴 ∪ (𝐵 ∩ 𝐶))
11	6, 10	eqeq12i 2783	. 2 ⊢ ((𝐶 ∩ (𝐵 ∪ 𝐴)) = ((𝐶 ∩ 𝐵) ∪ 𝐴) ↔ ((𝐴 ∪ 𝐵) ∩ 𝐶) = (𝐴 ∪ (𝐵 ∩ 𝐶)))
12	1, 2, 11	3bitri 300	1 ⊢ (𝐴 ⊆ 𝐶 ↔ ((𝐴 ∪ 𝐵) ∩ 𝐶) = (𝐴 ∪ (𝐵 ∩ 𝐶)))

Syntax hints:   ↔ wb 209   = wceq 1563   ∪ cun 3905   ∩ cin 3906   ⊆ wss 3907

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1566  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-rab 3418  df-v 3459  df-un 3912  df-in 3914  df-ss 3924

This theorem is referenced by: (None)