Theorem rp-fakeinunass 44039

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 44037	. . 3 ⊢ ((𝑥 ∈ 𝐶 → 𝑥 ∈ 𝐴) ↔ (((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∨ 𝑥 ∈ 𝐶) ↔ (𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐶))))
2	1	albii 1833	. 2 ⊢ (∀𝑥(𝑥 ∈ 𝐶 → 𝑥 ∈ 𝐴) ↔ ∀𝑥(((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∨ 𝑥 ∈ 𝐶) ↔ (𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐶))))
3		df-ss 3916	. 2 ⊢ (𝐶 ⊆ 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝐶 → 𝑥 ∈ 𝐴))
4		dfcleq 2749	. . 3 ⊢ (((𝐴 ∩ 𝐵) ∪ 𝐶) = (𝐴 ∩ (𝐵 ∪ 𝐶)) ↔ ∀𝑥(𝑥 ∈ ((𝐴 ∩ 𝐵) ∪ 𝐶) ↔ 𝑥 ∈ (𝐴 ∩ (𝐵 ∪ 𝐶))))
5		elun 4101	. . . . . 6 ⊢ (𝑥 ∈ ((𝐴 ∩ 𝐵) ∪ 𝐶) ↔ (𝑥 ∈ (𝐴 ∩ 𝐵) ∨ 𝑥 ∈ 𝐶))
6		elin 3915	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
7	6	orbi1i 922	. . . . . 6 ⊢ ((𝑥 ∈ (𝐴 ∩ 𝐵) ∨ 𝑥 ∈ 𝐶) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∨ 𝑥 ∈ 𝐶))
8	5, 7	bitri 277	. . . . 5 ⊢ (𝑥 ∈ ((𝐴 ∩ 𝐵) ∪ 𝐶) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∨ 𝑥 ∈ 𝐶))
9		elin 3915	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∩ (𝐵 ∪ 𝐶)) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 ∪ 𝐶)))
10		elun 4101	. . . . . . 7 ⊢ (𝑥 ∈ (𝐵 ∪ 𝐶) ↔ (𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐶))
11	10	anbi2i 631	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 ∪ 𝐶)) ↔ (𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐶)))
12	9, 11	bitri 277	. . . . 5 ⊢ (𝑥 ∈ (𝐴 ∩ (𝐵 ∪ 𝐶)) ↔ (𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐶)))
13	8, 12	bibi12i 341	. . . 4 ⊢ ((𝑥 ∈ ((𝐴 ∩ 𝐵) ∪ 𝐶) ↔ 𝑥 ∈ (𝐴 ∩ (𝐵 ∪ 𝐶))) ↔ (((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∨ 𝑥 ∈ 𝐶) ↔ (𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐶))))
14	13	albii 1833	. . 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 277	. 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 305	1 ⊢ (𝐶 ⊆ 𝐴 ↔ ((𝐴 ∩ 𝐵) ∪ 𝐶) = (𝐴 ∩ (𝐵 ∪ 𝐶)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∨ wo 856  ∀wal 1552   = wceq 1554   ∈ wcel 2136   ∪ cun 3897   ∩ cin 3898   ⊆ wss 3899

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-ext 2728

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-tru 1557  df-ex 1794  df-sb 2085  df-clab 2735  df-cleq 2748  df-clel 2831  df-v 3450  df-un 3904  df-in 3906  df-ss 3916

This theorem is referenced by:  rp-fakeuninass  44040