Theorem euoreqb 44601

Description: There is a set which is equal to one of two other sets iff the other sets are equal. (Contributed by AV, 24-Jan-2023.)

Assertion

Ref	Expression
euoreqb	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (∃!𝑥 ∈ 𝑉 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ↔ 𝐴 = 𝐵))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑉

Proof of Theorem euoreqb

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq1 2742	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝐴 ↔ 𝑦 = 𝐴))
2		eqeq1 2742	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝐵 ↔ 𝑦 = 𝐵))
3	1, 2	orbi12d 916	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ↔ (𝑦 = 𝐴 ∨ 𝑦 = 𝐵)))
4	3	reu8 3668	. . 3 ⊢ (∃!𝑥 ∈ 𝑉 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ↔ ∃𝑥 ∈ 𝑉 ((𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ∧ ∀𝑦 ∈ 𝑉 ((𝑦 = 𝐴 ∨ 𝑦 = 𝐵) → 𝑥 = 𝑦)))
5		simprlr 777	. . . . . . . . . 10 ⊢ ((𝑥 = 𝐴 ∧ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝑥 ∈ 𝑉)) → 𝐵 ∈ 𝑉)
6		eqeq1 2742	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐵 → (𝑦 = 𝐴 ↔ 𝐵 = 𝐴))
7		eqeq1 2742	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐵 → (𝑦 = 𝐵 ↔ 𝐵 = 𝐵))
8	6, 7	orbi12d 916	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝐵 → ((𝑦 = 𝐴 ∨ 𝑦 = 𝐵) ↔ (𝐵 = 𝐴 ∨ 𝐵 = 𝐵)))
9		eqeq2 2750	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝐵 → (𝑥 = 𝑦 ↔ 𝑥 = 𝐵))
10	8, 9	imbi12d 345	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐵 → (((𝑦 = 𝐴 ∨ 𝑦 = 𝐵) → 𝑥 = 𝑦) ↔ ((𝐵 = 𝐴 ∨ 𝐵 = 𝐵) → 𝑥 = 𝐵)))
11	10	rspcv 3557	. . . . . . . . . 10 ⊢ (𝐵 ∈ 𝑉 → (∀𝑦 ∈ 𝑉 ((𝑦 = 𝐴 ∨ 𝑦 = 𝐵) → 𝑥 = 𝑦) → ((𝐵 = 𝐴 ∨ 𝐵 = 𝐵) → 𝑥 = 𝐵)))
12	5, 11	syl 17	. . . . . . . . 9 ⊢ ((𝑥 = 𝐴 ∧ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝑥 ∈ 𝑉)) → (∀𝑦 ∈ 𝑉 ((𝑦 = 𝐴 ∨ 𝑦 = 𝐵) → 𝑥 = 𝑦) → ((𝐵 = 𝐴 ∨ 𝐵 = 𝐵) → 𝑥 = 𝐵)))
13		ioran 981	. . . . . . . . . . . 12 ⊢ (¬ (𝐵 = 𝐴 ∨ 𝐵 = 𝐵) ↔ (¬ 𝐵 = 𝐴 ∧ ¬ 𝐵 = 𝐵))
14		eqid 2738	. . . . . . . . . . . . 13 ⊢ 𝐵 = 𝐵
15	14	pm2.24i 150	. . . . . . . . . . . 12 ⊢ (¬ 𝐵 = 𝐵 → ((𝑥 = 𝐴 ∧ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝑥 ∈ 𝑉)) → 𝐴 = 𝐵))
16	13, 15	simplbiim 505	. . . . . . . . . . 11 ⊢ (¬ (𝐵 = 𝐴 ∨ 𝐵 = 𝐵) → ((𝑥 = 𝐴 ∧ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝑥 ∈ 𝑉)) → 𝐴 = 𝐵))
17		eqtr2 2762	. . . . . . . . . . . . . 14 ⊢ ((𝑥 = 𝐴 ∧ 𝑥 = 𝐵) → 𝐴 = 𝐵)
18	17	ancoms 459	. . . . . . . . . . . . 13 ⊢ ((𝑥 = 𝐵 ∧ 𝑥 = 𝐴) → 𝐴 = 𝐵)
19	18	a1d 25	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝐵 ∧ 𝑥 = 𝐴) → (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝑥 ∈ 𝑉) → 𝐴 = 𝐵))
20	19	expimpd 454	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐵 → ((𝑥 = 𝐴 ∧ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝑥 ∈ 𝑉)) → 𝐴 = 𝐵))
21	16, 20	ja 186	. . . . . . . . . 10 ⊢ (((𝐵 = 𝐴 ∨ 𝐵 = 𝐵) → 𝑥 = 𝐵) → ((𝑥 = 𝐴 ∧ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝑥 ∈ 𝑉)) → 𝐴 = 𝐵))
22	21	com12 32	. . . . . . . . 9 ⊢ ((𝑥 = 𝐴 ∧ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝑥 ∈ 𝑉)) → (((𝐵 = 𝐴 ∨ 𝐵 = 𝐵) → 𝑥 = 𝐵) → 𝐴 = 𝐵))
23	12, 22	syld 47	. . . . . . . 8 ⊢ ((𝑥 = 𝐴 ∧ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝑥 ∈ 𝑉)) → (∀𝑦 ∈ 𝑉 ((𝑦 = 𝐴 ∨ 𝑦 = 𝐵) → 𝑥 = 𝑦) → 𝐴 = 𝐵))
24	23	ex 413	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝑥 ∈ 𝑉) → (∀𝑦 ∈ 𝑉 ((𝑦 = 𝐴 ∨ 𝑦 = 𝐵) → 𝑥 = 𝑦) → 𝐴 = 𝐵)))
25		simprll 776	. . . . . . . . . 10 ⊢ ((𝑥 = 𝐵 ∧ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝑥 ∈ 𝑉)) → 𝐴 ∈ 𝑉)
26		eqeq1 2742	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐴 → (𝑦 = 𝐴 ↔ 𝐴 = 𝐴))
27		eqeq1 2742	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐴 → (𝑦 = 𝐵 ↔ 𝐴 = 𝐵))
28	26, 27	orbi12d 916	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝐴 → ((𝑦 = 𝐴 ∨ 𝑦 = 𝐵) ↔ (𝐴 = 𝐴 ∨ 𝐴 = 𝐵)))
29		eqeq2 2750	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝐴 → (𝑥 = 𝑦 ↔ 𝑥 = 𝐴))
30	28, 29	imbi12d 345	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐴 → (((𝑦 = 𝐴 ∨ 𝑦 = 𝐵) → 𝑥 = 𝑦) ↔ ((𝐴 = 𝐴 ∨ 𝐴 = 𝐵) → 𝑥 = 𝐴)))
31	30	rspcv 3557	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → (∀𝑦 ∈ 𝑉 ((𝑦 = 𝐴 ∨ 𝑦 = 𝐵) → 𝑥 = 𝑦) → ((𝐴 = 𝐴 ∨ 𝐴 = 𝐵) → 𝑥 = 𝐴)))
32	25, 31	syl 17	. . . . . . . . 9 ⊢ ((𝑥 = 𝐵 ∧ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝑥 ∈ 𝑉)) → (∀𝑦 ∈ 𝑉 ((𝑦 = 𝐴 ∨ 𝑦 = 𝐵) → 𝑥 = 𝑦) → ((𝐴 = 𝐴 ∨ 𝐴 = 𝐵) → 𝑥 = 𝐴)))
33		ioran 981	. . . . . . . . . . . 12 ⊢ (¬ (𝐴 = 𝐴 ∨ 𝐴 = 𝐵) ↔ (¬ 𝐴 = 𝐴 ∧ ¬ 𝐴 = 𝐵))
34		eqid 2738	. . . . . . . . . . . . . 14 ⊢ 𝐴 = 𝐴
35	34	pm2.24i 150	. . . . . . . . . . . . 13 ⊢ (¬ 𝐴 = 𝐴 → ((𝑥 = 𝐵 ∧ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝑥 ∈ 𝑉)) → 𝐴 = 𝐵))
36	35	adantr 481	. . . . . . . . . . . 12 ⊢ ((¬ 𝐴 = 𝐴 ∧ ¬ 𝐴 = 𝐵) → ((𝑥 = 𝐵 ∧ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝑥 ∈ 𝑉)) → 𝐴 = 𝐵))
37	33, 36	sylbi 216	. . . . . . . . . . 11 ⊢ (¬ (𝐴 = 𝐴 ∨ 𝐴 = 𝐵) → ((𝑥 = 𝐵 ∧ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝑥 ∈ 𝑉)) → 𝐴 = 𝐵))
38	17	a1d 25	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝐴 ∧ 𝑥 = 𝐵) → (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝑥 ∈ 𝑉) → 𝐴 = 𝐵))
39	38	expimpd 454	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐴 → ((𝑥 = 𝐵 ∧ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝑥 ∈ 𝑉)) → 𝐴 = 𝐵))
40	37, 39	ja 186	. . . . . . . . . 10 ⊢ (((𝐴 = 𝐴 ∨ 𝐴 = 𝐵) → 𝑥 = 𝐴) → ((𝑥 = 𝐵 ∧ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝑥 ∈ 𝑉)) → 𝐴 = 𝐵))
41	40	com12 32	. . . . . . . . 9 ⊢ ((𝑥 = 𝐵 ∧ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝑥 ∈ 𝑉)) → (((𝐴 = 𝐴 ∨ 𝐴 = 𝐵) → 𝑥 = 𝐴) → 𝐴 = 𝐵))
42	32, 41	syld 47	. . . . . . . 8 ⊢ ((𝑥 = 𝐵 ∧ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝑥 ∈ 𝑉)) → (∀𝑦 ∈ 𝑉 ((𝑦 = 𝐴 ∨ 𝑦 = 𝐵) → 𝑥 = 𝑦) → 𝐴 = 𝐵))
43	42	ex 413	. . . . . . 7 ⊢ (𝑥 = 𝐵 → (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝑥 ∈ 𝑉) → (∀𝑦 ∈ 𝑉 ((𝑦 = 𝐴 ∨ 𝑦 = 𝐵) → 𝑥 = 𝑦) → 𝐴 = 𝐵)))
44	24, 43	jaoi 854	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝑥 ∈ 𝑉) → (∀𝑦 ∈ 𝑉 ((𝑦 = 𝐴 ∨ 𝑦 = 𝐵) → 𝑥 = 𝑦) → 𝐴 = 𝐵)))
45	44	com12 32	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝑥 ∈ 𝑉) → ((𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → (∀𝑦 ∈ 𝑉 ((𝑦 = 𝐴 ∨ 𝑦 = 𝐵) → 𝑥 = 𝑦) → 𝐴 = 𝐵)))
46	45	impd 411	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝑥 ∈ 𝑉) → (((𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ∧ ∀𝑦 ∈ 𝑉 ((𝑦 = 𝐴 ∨ 𝑦 = 𝐵) → 𝑥 = 𝑦)) → 𝐴 = 𝐵))
47	46	rexlimdva 3213	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (∃𝑥 ∈ 𝑉 ((𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ∧ ∀𝑦 ∈ 𝑉 ((𝑦 = 𝐴 ∨ 𝑦 = 𝐵) → 𝑥 = 𝑦)) → 𝐴 = 𝐵))
48	4, 47	syl5bi 241	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (∃!𝑥 ∈ 𝑉 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → 𝐴 = 𝐵))
49		reueq 3672	. . . . . . 7 ⊢ (𝐵 ∈ 𝑉 ↔ ∃!𝑥 ∈ 𝑉 𝑥 = 𝐵)
50	49	biimpi 215	. . . . . 6 ⊢ (𝐵 ∈ 𝑉 → ∃!𝑥 ∈ 𝑉 𝑥 = 𝐵)
51	50	adantl 482	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ∃!𝑥 ∈ 𝑉 𝑥 = 𝐵)
52	51	adantr 481	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝐴 = 𝐵) → ∃!𝑥 ∈ 𝑉 𝑥 = 𝐵)
53		eqeq2 2750	. . . . . . . 8 ⊢ (𝐴 = 𝐵 → (𝑥 = 𝐴 ↔ 𝑥 = 𝐵))
54	53	adantl 482	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝐴 = 𝐵) → (𝑥 = 𝐴 ↔ 𝑥 = 𝐵))
55	54	orbi1d 914	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝐴 = 𝐵) → ((𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ↔ (𝑥 = 𝐵 ∨ 𝑥 = 𝐵)))
56		oridm 902	. . . . . 6 ⊢ ((𝑥 = 𝐵 ∨ 𝑥 = 𝐵) ↔ 𝑥 = 𝐵)
57	55, 56	bitrdi 287	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝐴 = 𝐵) → ((𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ↔ 𝑥 = 𝐵))
58	57	reubidv 3323	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝐴 = 𝐵) → (∃!𝑥 ∈ 𝑉 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ↔ ∃!𝑥 ∈ 𝑉 𝑥 = 𝐵))
59	52, 58	mpbird 256	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝐴 = 𝐵) → ∃!𝑥 ∈ 𝑉 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵))
60	59	ex 413	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴 = 𝐵 → ∃!𝑥 ∈ 𝑉 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵)))
61	48, 60	impbid 211	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (∃!𝑥 ∈ 𝑉 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ↔ 𝐴 = 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 844   = wceq 1539   ∈ wcel 2106  ∀wral 3064  ∃wrex 3065  ∃!wreu 3066

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072

This theorem is referenced by:  quad1  45072  requad1  45074