Theorem rp-fakeinunass 40223

Description: A special case where a mixture of intersection and union appears to conform to a mixed associative law. (Contributed by RP, 26-Feb-2020.)

Assertion

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

Proof of Theorem rp-fakeinunass

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rp-fakeanorass 40221	. . 3 ⊢ ((𝑥 ∈ 𝐶 → 𝑥 ∈ 𝐴) ↔ (((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∨ 𝑥 ∈ 𝐶) ↔ (𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐶))))
2	1	albii 1821	. 2 ⊢ (∀𝑥(𝑥 ∈ 𝐶 → 𝑥 ∈ 𝐴) ↔ ∀𝑥(((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∨ 𝑥 ∈ 𝐶) ↔ (𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐶))))
3		dfss2 3901	. 2 ⊢ (𝐶 ⊆ 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝐶 → 𝑥 ∈ 𝐴))
4		dfcleq 2792	. . 3 ⊢ (((𝐴 ∩ 𝐵) ∪ 𝐶) = (𝐴 ∩ (𝐵 ∪ 𝐶)) ↔ ∀𝑥(𝑥 ∈ ((𝐴 ∩ 𝐵) ∪ 𝐶) ↔ 𝑥 ∈ (𝐴 ∩ (𝐵 ∪ 𝐶))))
5		elun 4076	. . . . . 6 ⊢ (𝑥 ∈ ((𝐴 ∩ 𝐵) ∪ 𝐶) ↔ (𝑥 ∈ (𝐴 ∩ 𝐵) ∨ 𝑥 ∈ 𝐶))
6		elin 3897	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
7	6	orbi1i 911	. . . . . 6 ⊢ ((𝑥 ∈ (𝐴 ∩ 𝐵) ∨ 𝑥 ∈ 𝐶) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∨ 𝑥 ∈ 𝐶))
8	5, 7	bitri 278	. . . . 5 ⊢ (𝑥 ∈ ((𝐴 ∩ 𝐵) ∪ 𝐶) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∨ 𝑥 ∈ 𝐶))
9		elin 3897	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∩ (𝐵 ∪ 𝐶)) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 ∪ 𝐶)))
10		elun 4076	. . . . . . 7 ⊢ (𝑥 ∈ (𝐵 ∪ 𝐶) ↔ (𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐶))
11	10	anbi2i 625	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 ∪ 𝐶)) ↔ (𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐶)))
12	9, 11	bitri 278	. . . . 5 ⊢ (𝑥 ∈ (𝐴 ∩ (𝐵 ∪ 𝐶)) ↔ (𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐶)))
13	8, 12	bibi12i 343	. . . 4 ⊢ ((𝑥 ∈ ((𝐴 ∩ 𝐵) ∪ 𝐶) ↔ 𝑥 ∈ (𝐴 ∩ (𝐵 ∪ 𝐶))) ↔ (((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∨ 𝑥 ∈ 𝐶) ↔ (𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐶))))
14	13	albii 1821	. . 3 ⊢ (∀𝑥(𝑥 ∈ ((𝐴 ∩ 𝐵) ∪ 𝐶) ↔ 𝑥 ∈ (𝐴 ∩ (𝐵 ∪ 𝐶))) ↔ ∀𝑥(((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∨ 𝑥 ∈ 𝐶) ↔ (𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐶))))
15	4, 14	bitri 278	. 2 ⊢ (((𝐴 ∩ 𝐵) ∪ 𝐶) = (𝐴 ∩ (𝐵 ∪ 𝐶)) ↔ ∀𝑥(((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∨ 𝑥 ∈ 𝐶) ↔ (𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐶))))
16	2, 3, 15	3bitr4i 306	1 ⊢ (𝐶 ⊆ 𝐴 ↔ ((𝐴 ∩ 𝐵) ∪ 𝐶) = (𝐴 ∩ (𝐵 ∪ 𝐶)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844  ∀wal 1536   = wceq 1538   ∈ wcel 2111   ∪ cun 3879   ∩ cin 3880   ⊆ wss 3881

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443  df-un 3886  df-in 3888  df-ss 3898

This theorem is referenced by:  rp-fakeuninass  40224