Theorem unineq 4234

Description: Infer equality from equalities of union and intersection. Exercise 20 of [Enderton] p. 32 and its converse. (Contributed by NM, 10-Aug-2004.)

Assertion

Ref	Expression
unineq	⊢ (((𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐶) ∧ (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐶)) ↔ 𝐴 = 𝐵)

Proof of Theorem unineq

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq2 2850	. . . . . . 7 ⊢ ((𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐶) → (𝑥 ∈ (𝐴 ∩ 𝐶) ↔ 𝑥 ∈ (𝐵 ∩ 𝐶)))
2		elin 3915	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐶) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐶))
3		elin 3915	. . . . . . 7 ⊢ (𝑥 ∈ (𝐵 ∩ 𝐶) ↔ (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶))
4	1, 2, 3	3bitr3g 316	. . . . . 6 ⊢ ((𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐶) → ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐶) ↔ (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶)))
5		iba 537	. . . . . . 7 ⊢ (𝑥 ∈ 𝐶 → (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐶)))
6		iba 537	. . . . . . 7 ⊢ (𝑥 ∈ 𝐶 → (𝑥 ∈ 𝐵 ↔ (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶)))
7	5, 6	bibi12d 348	. . . . . 6 ⊢ (𝑥 ∈ 𝐶 → ((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐶) ↔ (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶))))
8	4, 7	imbitrrid 249	. . . . 5 ⊢ (𝑥 ∈ 𝐶 → ((𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐶) → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)))
9	8	adantld 496	. . . 4 ⊢ (𝑥 ∈ 𝐶 → (((𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐶) ∧ (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐶)) → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)))
10		uncom 4105	. . . . . . . . 9 ⊢ (𝐴 ∪ 𝐶) = (𝐶 ∪ 𝐴)
11		uncom 4105	. . . . . . . . 9 ⊢ (𝐵 ∪ 𝐶) = (𝐶 ∪ 𝐵)
12	10, 11	eqeq12i 2779	. . . . . . . 8 ⊢ ((𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐶) ↔ (𝐶 ∪ 𝐴) = (𝐶 ∪ 𝐵))
13		eleq2 2850	. . . . . . . 8 ⊢ ((𝐶 ∪ 𝐴) = (𝐶 ∪ 𝐵) → (𝑥 ∈ (𝐶 ∪ 𝐴) ↔ 𝑥 ∈ (𝐶 ∪ 𝐵)))
14	12, 13	sylbi 220	. . . . . . 7 ⊢ ((𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐶) → (𝑥 ∈ (𝐶 ∪ 𝐴) ↔ 𝑥 ∈ (𝐶 ∪ 𝐵)))
15		elun 4100	. . . . . . 7 ⊢ (𝑥 ∈ (𝐶 ∪ 𝐴) ↔ (𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐴))
16		elun 4100	. . . . . . 7 ⊢ (𝑥 ∈ (𝐶 ∪ 𝐵) ↔ (𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐵))
17	14, 15, 16	3bitr3g 316	. . . . . 6 ⊢ ((𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐶) → ((𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐴) ↔ (𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐵)))
18		biorf 950	. . . . . . 7 ⊢ (¬ 𝑥 ∈ 𝐶 → (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐴)))
19		biorf 950	. . . . . . 7 ⊢ (¬ 𝑥 ∈ 𝐶 → (𝑥 ∈ 𝐵 ↔ (𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐵)))
20	18, 19	bibi12d 348	. . . . . 6 ⊢ (¬ 𝑥 ∈ 𝐶 → ((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ↔ ((𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐴) ↔ (𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐵))))
21	17, 20	imbitrrid 249	. . . . 5 ⊢ (¬ 𝑥 ∈ 𝐶 → ((𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐶) → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)))
22	21	adantrd 497	. . . 4 ⊢ (¬ 𝑥 ∈ 𝐶 → (((𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐶) ∧ (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐶)) → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)))
23	9, 22	pm2.61i 184	. . 3 ⊢ (((𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐶) ∧ (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐶)) → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
24	23	eqrdv 2759	. 2 ⊢ (((𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐶) ∧ (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐶)) → 𝐴 = 𝐵)
25		uneq1 4108	. . 3 ⊢ (𝐴 = 𝐵 → (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐶))
26		ineq1 4159	. . 3 ⊢ (𝐴 = 𝐵 → (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐶))
27	25, 26	jca 521	. 2 ⊢ (𝐴 = 𝐵 → ((𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐶) ∧ (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐶)))
28	24, 27	impbii 212	1 ⊢ (((𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐶) ∧ (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐶)) ↔ 𝐴 = 𝐵)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2145   ∪ cun 3897   ∩ cin 3898

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-rab 3414  df-v 3453  df-un 3904  df-in 3906

This theorem is used by: (None)