Theorem rp-fakeinunass 40676

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 40674	. . 3 ⊢ ((𝑥 ∈ 𝐶 → 𝑥 ∈ 𝐴) ↔ (((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∨ 𝑥 ∈ 𝐶) ↔ (𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐶))))
2	1	albii 1826	. 2 ⊢ (∀𝑥(𝑥 ∈ 𝐶 → 𝑥 ∈ 𝐴) ↔ ∀𝑥(((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∨ 𝑥 ∈ 𝐶) ↔ (𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐶))))
3		dfss2 3863	. 2 ⊢ (𝐶 ⊆ 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝐶 → 𝑥 ∈ 𝐴))
4		dfcleq 2731	. . 3 ⊢ (((𝐴 ∩ 𝐵) ∪ 𝐶) = (𝐴 ∩ (𝐵 ∪ 𝐶)) ↔ ∀𝑥(𝑥 ∈ ((𝐴 ∩ 𝐵) ∪ 𝐶) ↔ 𝑥 ∈ (𝐴 ∩ (𝐵 ∪ 𝐶))))
5		elun 4039	. . . . . 6 ⊢ (𝑥 ∈ ((𝐴 ∩ 𝐵) ∪ 𝐶) ↔ (𝑥 ∈ (𝐴 ∩ 𝐵) ∨ 𝑥 ∈ 𝐶))
6		elin 3859	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
7	6	orbi1i 913	. . . . . 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 3859	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∩ (𝐵 ∪ 𝐶)) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 ∪ 𝐶)))
10		elun 4039	. . . . . . 7 ⊢ (𝑥 ∈ (𝐵 ∪ 𝐶) ↔ (𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐶))
11	10	anbi2i 626	. . . . . 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 1826	. . 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 846  ∀wal 1540   = wceq 1542   ∈ wcel 2114   ∪ cun 3841   ∩ cin 3842   ⊆ wss 3843

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2710

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-tru 1545  df-ex 1787  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-v 3400  df-un 3848  df-in 3850  df-ss 3860

This theorem is referenced by:  rp-fakeuninass  40677