Theorem paireqne 42442

Description: Two sets are not equal iff there is exactly one proper pair whose elements are either one of these sets. (Contributed by AV, 27-Jan-2023.)

Hypotheses

Ref	Expression
paireqne.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
paireqne.b	⊢ (𝜑 → 𝐵 ∈ 𝑉)
paireqne.p	⊢ 𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}

Assertion

Ref	Expression
paireqne	⊢ (𝜑 → (∃!𝑝 ∈ 𝑃 ∀𝑥 ∈ 𝑝 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ↔ 𝐴 ≠ 𝐵))

Distinct variable groups:   𝐴,𝑝,𝑥   𝐵,𝑝,𝑥   𝑃,𝑝,𝑥   𝑥,𝑉   𝜑,𝑝,𝑥

Allowed substitution hint:   𝑉(𝑝)

Proof of Theorem paireqne

Dummy variables 𝑎 𝑏 𝑞 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		raleq 3329	. . . 4 ⊢ (𝑝 = 𝑞 → (∀𝑥 ∈ 𝑝 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ↔ ∀𝑥 ∈ 𝑞 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵)))
2	1	reu8 3613	. . 3 ⊢ (∃!𝑝 ∈ 𝑃 ∀𝑥 ∈ 𝑝 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ↔ ∃𝑝 ∈ 𝑃 (∀𝑥 ∈ 𝑝 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ∧ ∀𝑞 ∈ 𝑃 (∀𝑥 ∈ 𝑞 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → 𝑝 = 𝑞)))
3		paireqne.p	. . . . . . . 8 ⊢ 𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}
4	3	eleq2i 2850	. . . . . . 7 ⊢ (𝑝 ∈ 𝑃 ↔ 𝑝 ∈ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2})
5		elss2prb 13583	. . . . . . 7 ⊢ (𝑝 ∈ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} ↔ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏}))
6	4, 5	bitri 267	. . . . . 6 ⊢ (𝑝 ∈ 𝑃 ↔ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏}))
7		raleq 3329	. . . . . . . . . . . 12 ⊢ (𝑝 = {𝑎, 𝑏} → (∀𝑥 ∈ 𝑝 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ↔ ∀𝑥 ∈ {𝑎, 𝑏} (𝑥 = 𝐴 ∨ 𝑥 = 𝐵)))
8		vex 3400	. . . . . . . . . . . . 13 ⊢ 𝑎 ∈ V
9		vex 3400	. . . . . . . . . . . . 13 ⊢ 𝑏 ∈ V
10		eqeq1 2781	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑎 → (𝑥 = 𝐴 ↔ 𝑎 = 𝐴))
11		eqeq1 2781	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑎 → (𝑥 = 𝐵 ↔ 𝑎 = 𝐵))
12	10, 11	orbi12d 905	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑎 → ((𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ↔ (𝑎 = 𝐴 ∨ 𝑎 = 𝐵)))
13		eqeq1 2781	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑏 → (𝑥 = 𝐴 ↔ 𝑏 = 𝐴))
14		eqeq1 2781	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑏 → (𝑥 = 𝐵 ↔ 𝑏 = 𝐵))
15	13, 14	orbi12d 905	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑏 → ((𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ↔ (𝑏 = 𝐴 ∨ 𝑏 = 𝐵)))
16	8, 9, 12, 15	ralpr 4466	. . . . . . . . . . . 12 ⊢ (∀𝑥 ∈ {𝑎, 𝑏} (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ↔ ((𝑎 = 𝐴 ∨ 𝑎 = 𝐵) ∧ (𝑏 = 𝐴 ∨ 𝑏 = 𝐵)))
17	7, 16	syl6bb 279	. . . . . . . . . . 11 ⊢ (𝑝 = {𝑎, 𝑏} → (∀𝑥 ∈ 𝑝 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ↔ ((𝑎 = 𝐴 ∨ 𝑎 = 𝐵) ∧ (𝑏 = 𝐴 ∨ 𝑏 = 𝐵))))
18		eqeq1 2781	. . . . . . . . . . . . 13 ⊢ (𝑝 = {𝑎, 𝑏} → (𝑝 = 𝑞 ↔ {𝑎, 𝑏} = 𝑞))
19	18	imbi2d 332	. . . . . . . . . . . 12 ⊢ (𝑝 = {𝑎, 𝑏} → ((∀𝑥 ∈ 𝑞 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → 𝑝 = 𝑞) ↔ (∀𝑥 ∈ 𝑞 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → {𝑎, 𝑏} = 𝑞)))
20	19	ralbidv 3167	. . . . . . . . . . 11 ⊢ (𝑝 = {𝑎, 𝑏} → (∀𝑞 ∈ 𝑃 (∀𝑥 ∈ 𝑞 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → 𝑝 = 𝑞) ↔ ∀𝑞 ∈ 𝑃 (∀𝑥 ∈ 𝑞 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → {𝑎, 𝑏} = 𝑞)))
21	17, 20	anbi12d 624	. . . . . . . . . 10 ⊢ (𝑝 = {𝑎, 𝑏} → ((∀𝑥 ∈ 𝑝 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ∧ ∀𝑞 ∈ 𝑃 (∀𝑥 ∈ 𝑞 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → 𝑝 = 𝑞)) ↔ (((𝑎 = 𝐴 ∨ 𝑎 = 𝐵) ∧ (𝑏 = 𝐴 ∨ 𝑏 = 𝐵)) ∧ ∀𝑞 ∈ 𝑃 (∀𝑥 ∈ 𝑞 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → {𝑎, 𝑏} = 𝑞))))
22	21	ad2antll 719	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})) → ((∀𝑥 ∈ 𝑝 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ∧ ∀𝑞 ∈ 𝑃 (∀𝑥 ∈ 𝑞 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → 𝑝 = 𝑞)) ↔ (((𝑎 = 𝐴 ∨ 𝑎 = 𝐵) ∧ (𝑏 = 𝐴 ∨ 𝑏 = 𝐵)) ∧ ∀𝑞 ∈ 𝑃 (∀𝑥 ∈ 𝑞 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → {𝑎, 𝑏} = 𝑞))))
23		paireqne.a	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
24		paireqne.b	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
25	23, 24	jca 507	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉))
26		prelpwi 5147	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → {𝐴, 𝐵} ∈ 𝒫 𝑉)
27	25, 26	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → {𝐴, 𝐵} ∈ 𝒫 𝑉)
28	27	ad3antrrr 720	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})) ∧ ((𝑎 = 𝐴 ∨ 𝑎 = 𝐵) ∧ (𝑏 = 𝐴 ∨ 𝑏 = 𝐵))) → {𝐴, 𝐵} ∈ 𝒫 𝑉)
29		hashprg 13497	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → (𝑎 ≠ 𝑏 ↔ (♯‘{𝑎, 𝑏}) = 2))
30	29	adantl 475	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → (𝑎 ≠ 𝑏 ↔ (♯‘{𝑎, 𝑏}) = 2))
31	30	biimpd 221	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → (𝑎 ≠ 𝑏 → (♯‘{𝑎, 𝑏}) = 2))
32	31	com12 32	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 ≠ 𝑏 → ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → (♯‘{𝑎, 𝑏}) = 2))
33	32	adantr 474	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏}) → ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → (♯‘{𝑎, 𝑏}) = 2))
34	33	impcom 398	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})) → (♯‘{𝑎, 𝑏}) = 2)
35	34	adantr 474	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})) ∧ ((𝑎 = 𝐴 ∨ 𝑎 = 𝐵) ∧ (𝑏 = 𝐴 ∨ 𝑏 = 𝐵))) → (♯‘{𝑎, 𝑏}) = 2)
36		eqtr3 2800	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑏 = 𝐴 ∧ 𝑎 = 𝐴) → 𝑏 = 𝑎)
37		eqneqall 2979	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑎 = 𝑏 → (𝑎 ≠ 𝑏 → (𝑝 = {𝑎, 𝑏} → {𝐴, 𝐵} = {𝑎, 𝑏})))
38	37	impd 400	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑎 = 𝑏 → ((𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏}) → {𝐴, 𝐵} = {𝑎, 𝑏}))
39	38	a1d 25	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑎 = 𝑏 → ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ((𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏}) → {𝐴, 𝐵} = {𝑎, 𝑏})))
40	39	impd 400	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑎 = 𝑏 → (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})) → {𝐴, 𝐵} = {𝑎, 𝑏}))
41	40	equcoms 2066	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑏 = 𝑎 → (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})) → {𝐴, 𝐵} = {𝑎, 𝑏}))
42	36, 41	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑏 = 𝐴 ∧ 𝑎 = 𝐴) → (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})) → {𝐴, 𝐵} = {𝑎, 𝑏}))
43	42	ex 403	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑏 = 𝐴 → (𝑎 = 𝐴 → (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})) → {𝐴, 𝐵} = {𝑎, 𝑏})))
44		preq12 4501	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → {𝑎, 𝑏} = {𝐴, 𝐵})
45	44	eqcomd 2783	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → {𝐴, 𝐵} = {𝑎, 𝑏})
46	45	a1d 25	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})) → {𝐴, 𝐵} = {𝑎, 𝑏}))
47	46	expcom 404	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑏 = 𝐵 → (𝑎 = 𝐴 → (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})) → {𝐴, 𝐵} = {𝑎, 𝑏})))
48	43, 47	jaoi 846	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑏 = 𝐴 ∨ 𝑏 = 𝐵) → (𝑎 = 𝐴 → (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})) → {𝐴, 𝐵} = {𝑎, 𝑏})))
49	48	com12 32	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = 𝐴 → ((𝑏 = 𝐴 ∨ 𝑏 = 𝐵) → (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})) → {𝐴, 𝐵} = {𝑎, 𝑏})))
50		prcom 4498	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ {𝑎, 𝑏} = {𝑏, 𝑎}
51		preq12 4501	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑏 = 𝐴 ∧ 𝑎 = 𝐵) → {𝑏, 𝑎} = {𝐴, 𝐵})
52	50, 51	syl5eq 2825	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑏 = 𝐴 ∧ 𝑎 = 𝐵) → {𝑎, 𝑏} = {𝐴, 𝐵})
53	52	eqcomd 2783	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑏 = 𝐴 ∧ 𝑎 = 𝐵) → {𝐴, 𝐵} = {𝑎, 𝑏})
54	53	a1d 25	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑏 = 𝐴 ∧ 𝑎 = 𝐵) → (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})) → {𝐴, 𝐵} = {𝑎, 𝑏}))
55	54	ex 403	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑏 = 𝐴 → (𝑎 = 𝐵 → (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})) → {𝐴, 𝐵} = {𝑎, 𝑏})))
56		eqtr3 2800	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑏 = 𝐵 ∧ 𝑎 = 𝐵) → 𝑏 = 𝑎)
57	56, 41	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑏 = 𝐵 ∧ 𝑎 = 𝐵) → (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})) → {𝐴, 𝐵} = {𝑎, 𝑏}))
58	57	ex 403	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑏 = 𝐵 → (𝑎 = 𝐵 → (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})) → {𝐴, 𝐵} = {𝑎, 𝑏})))
59	55, 58	jaoi 846	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑏 = 𝐴 ∨ 𝑏 = 𝐵) → (𝑎 = 𝐵 → (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})) → {𝐴, 𝐵} = {𝑎, 𝑏})))
60	59	com12 32	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = 𝐵 → ((𝑏 = 𝐴 ∨ 𝑏 = 𝐵) → (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})) → {𝐴, 𝐵} = {𝑎, 𝑏})))
61	49, 60	jaoi 846	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑎 = 𝐴 ∨ 𝑎 = 𝐵) → ((𝑏 = 𝐴 ∨ 𝑏 = 𝐵) → (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})) → {𝐴, 𝐵} = {𝑎, 𝑏})))
62	61	imp 397	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑎 = 𝐴 ∨ 𝑎 = 𝐵) ∧ (𝑏 = 𝐴 ∨ 𝑏 = 𝐵)) → (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})) → {𝐴, 𝐵} = {𝑎, 𝑏}))
63	62	impcom 398	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})) ∧ ((𝑎 = 𝐴 ∨ 𝑎 = 𝐵) ∧ (𝑏 = 𝐴 ∨ 𝑏 = 𝐵))) → {𝐴, 𝐵} = {𝑎, 𝑏})
64	63	fveqeq2d 6454	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})) ∧ ((𝑎 = 𝐴 ∨ 𝑎 = 𝐵) ∧ (𝑏 = 𝐴 ∨ 𝑏 = 𝐵))) → ((♯‘{𝐴, 𝐵}) = 2 ↔ (♯‘{𝑎, 𝑏}) = 2))
65	35, 64	mpbird 249	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})) ∧ ((𝑎 = 𝐴 ∨ 𝑎 = 𝐵) ∧ (𝑏 = 𝐴 ∨ 𝑏 = 𝐵))) → (♯‘{𝐴, 𝐵}) = 2)
66	28, 65	jca 507	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})) ∧ ((𝑎 = 𝐴 ∨ 𝑎 = 𝐵) ∧ (𝑏 = 𝐴 ∨ 𝑏 = 𝐵))) → ({𝐴, 𝐵} ∈ 𝒫 𝑉 ∧ (♯‘{𝐴, 𝐵}) = 2))
67	3	eleq2i 2850	. . . . . . . . . . . . . . 15 ⊢ ({𝐴, 𝐵} ∈ 𝑃 ↔ {𝐴, 𝐵} ∈ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2})
68		fveqeq2 6455	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = {𝐴, 𝐵} → ((♯‘𝑥) = 2 ↔ (♯‘{𝐴, 𝐵}) = 2))
69	68	elrab 3571	. . . . . . . . . . . . . . 15 ⊢ ({𝐴, 𝐵} ∈ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} ↔ ({𝐴, 𝐵} ∈ 𝒫 𝑉 ∧ (♯‘{𝐴, 𝐵}) = 2))
70	67, 69	bitri 267	. . . . . . . . . . . . . 14 ⊢ ({𝐴, 𝐵} ∈ 𝑃 ↔ ({𝐴, 𝐵} ∈ 𝒫 𝑉 ∧ (♯‘{𝐴, 𝐵}) = 2))
71	66, 70	sylibr 226	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})) ∧ ((𝑎 = 𝐴 ∨ 𝑎 = 𝐵) ∧ (𝑏 = 𝐴 ∨ 𝑏 = 𝐵))) → {𝐴, 𝐵} ∈ 𝑃)
72		raleq 3329	. . . . . . . . . . . . . . 15 ⊢ (𝑞 = {𝐴, 𝐵} → (∀𝑥 ∈ 𝑞 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ↔ ∀𝑥 ∈ {𝐴, 𝐵} (𝑥 = 𝐴 ∨ 𝑥 = 𝐵)))
73		eqeq2 2788	. . . . . . . . . . . . . . 15 ⊢ (𝑞 = {𝐴, 𝐵} → ({𝑎, 𝑏} = 𝑞 ↔ {𝑎, 𝑏} = {𝐴, 𝐵}))
74	72, 73	imbi12d 336	. . . . . . . . . . . . . 14 ⊢ (𝑞 = {𝐴, 𝐵} → ((∀𝑥 ∈ 𝑞 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → {𝑎, 𝑏} = 𝑞) ↔ (∀𝑥 ∈ {𝐴, 𝐵} (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → {𝑎, 𝑏} = {𝐴, 𝐵})))
75	74	rspcv 3506	. . . . . . . . . . . . 13 ⊢ ({𝐴, 𝐵} ∈ 𝑃 → (∀𝑞 ∈ 𝑃 (∀𝑥 ∈ 𝑞 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → {𝑎, 𝑏} = 𝑞) → (∀𝑥 ∈ {𝐴, 𝐵} (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → {𝑎, 𝑏} = {𝐴, 𝐵})))
76	71, 75	syl 17	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})) ∧ ((𝑎 = 𝐴 ∨ 𝑎 = 𝐵) ∧ (𝑏 = 𝐴 ∨ 𝑏 = 𝐵))) → (∀𝑞 ∈ 𝑃 (∀𝑥 ∈ 𝑞 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → {𝑎, 𝑏} = 𝑞) → (∀𝑥 ∈ {𝐴, 𝐵} (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → {𝑎, 𝑏} = {𝐴, 𝐵})))
77		eqeq1 2781	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝐴 ↔ 𝐴 = 𝐴))
78		eqeq1 2781	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝐵 ↔ 𝐴 = 𝐵))
79	77, 78	orbi12d 905	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝐴 → ((𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ↔ (𝐴 = 𝐴 ∨ 𝐴 = 𝐵)))
80		eqeq1 2781	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝐵 → (𝑥 = 𝐴 ↔ 𝐵 = 𝐴))
81		eqeq1 2781	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝐵 → (𝑥 = 𝐵 ↔ 𝐵 = 𝐵))
82	80, 81	orbi12d 905	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝐵 → ((𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ↔ (𝐵 = 𝐴 ∨ 𝐵 = 𝐵)))
83	79, 82	ralprg 4462	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (∀𝑥 ∈ {𝐴, 𝐵} (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ↔ ((𝐴 = 𝐴 ∨ 𝐴 = 𝐵) ∧ (𝐵 = 𝐴 ∨ 𝐵 = 𝐵))))
84	25, 83	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (∀𝑥 ∈ {𝐴, 𝐵} (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ↔ ((𝐴 = 𝐴 ∨ 𝐴 = 𝐵) ∧ (𝐵 = 𝐴 ∨ 𝐵 = 𝐵))))
85	84	imbi1d 333	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((∀𝑥 ∈ {𝐴, 𝐵} (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → {𝑎, 𝑏} = {𝐴, 𝐵}) ↔ (((𝐴 = 𝐴 ∨ 𝐴 = 𝐵) ∧ (𝐵 = 𝐴 ∨ 𝐵 = 𝐵)) → {𝑎, 𝑏} = {𝐴, 𝐵})))
86	85	ad3antrrr 720	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})) ∧ ((𝑎 = 𝐴 ∨ 𝑎 = 𝐵) ∧ (𝑏 = 𝐴 ∨ 𝑏 = 𝐵))) → ((∀𝑥 ∈ {𝐴, 𝐵} (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → {𝑎, 𝑏} = {𝐴, 𝐵}) ↔ (((𝐴 = 𝐴 ∨ 𝐴 = 𝐵) ∧ (𝐵 = 𝐴 ∨ 𝐵 = 𝐵)) → {𝑎, 𝑏} = {𝐴, 𝐵})))
87		eqid 2777	. . . . . . . . . . . . . . . 16 ⊢ 𝐴 = 𝐴
88	87	orci 854	. . . . . . . . . . . . . . 15 ⊢ (𝐴 = 𝐴 ∨ 𝐴 = 𝐵)
89		eqid 2777	. . . . . . . . . . . . . . . 16 ⊢ 𝐵 = 𝐵
90	89	olci 855	. . . . . . . . . . . . . . 15 ⊢ (𝐵 = 𝐴 ∨ 𝐵 = 𝐵)
91		pm5.5 353	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 = 𝐴 ∨ 𝐴 = 𝐵) ∧ (𝐵 = 𝐴 ∨ 𝐵 = 𝐵)) → ((((𝐴 = 𝐴 ∨ 𝐴 = 𝐵) ∧ (𝐵 = 𝐴 ∨ 𝐵 = 𝐵)) → {𝑎, 𝑏} = {𝐴, 𝐵}) ↔ {𝑎, 𝑏} = {𝐴, 𝐵}))
92	88, 90, 91	mp2an 682	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 = 𝐴 ∨ 𝐴 = 𝐵) ∧ (𝐵 = 𝐴 ∨ 𝐵 = 𝐵)) → {𝑎, 𝑏} = {𝐴, 𝐵}) ↔ {𝑎, 𝑏} = {𝐴, 𝐵})
93	8, 9	pm3.2i 464	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 ∈ V ∧ 𝑏 ∈ V)
94		preq12bg 4614	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑎 ∈ V ∧ 𝑏 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → ({𝑎, 𝑏} = {𝐴, 𝐵} ↔ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) ∨ (𝑎 = 𝐵 ∧ 𝑏 = 𝐴))))
95	93, 25, 94	sylancr 581	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ({𝑎, 𝑏} = {𝐴, 𝐵} ↔ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) ∨ (𝑎 = 𝐵 ∧ 𝑏 = 𝐴))))
96	95	adantr 474	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ({𝑎, 𝑏} = {𝐴, 𝐵} ↔ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) ∨ (𝑎 = 𝐵 ∧ 𝑏 = 𝐴))))
97	96	adantr 474	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})) → ({𝑎, 𝑏} = {𝐴, 𝐵} ↔ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) ∨ (𝑎 = 𝐵 ∧ 𝑏 = 𝐴))))
98		eqeq12 2790	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (𝑎 = 𝑏 ↔ 𝐴 = 𝐵))
99	98	necon3bid 3012	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (𝑎 ≠ 𝑏 ↔ 𝐴 ≠ 𝐵))
100	99	biimpd 221	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (𝑎 ≠ 𝑏 → 𝐴 ≠ 𝐵))
101		eqeq12 2790	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑎 = 𝐵 ∧ 𝑏 = 𝐴) → (𝑎 = 𝑏 ↔ 𝐵 = 𝐴))
102	101	necon3bid 3012	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑎 = 𝐵 ∧ 𝑏 = 𝐴) → (𝑎 ≠ 𝑏 ↔ 𝐵 ≠ 𝐴))
103	102	biimpd 221	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑎 = 𝐵 ∧ 𝑏 = 𝐴) → (𝑎 ≠ 𝑏 → 𝐵 ≠ 𝐴))
104		necom 3021	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐴 ≠ 𝐵 ↔ 𝐵 ≠ 𝐴)
105	103, 104	syl6ibr 244	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑎 = 𝐵 ∧ 𝑏 = 𝐴) → (𝑎 ≠ 𝑏 → 𝐴 ≠ 𝐵))
106	100, 105	jaoi 846	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) ∨ (𝑎 = 𝐵 ∧ 𝑏 = 𝐴)) → (𝑎 ≠ 𝑏 → 𝐴 ≠ 𝐵))
107	106	com12 32	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 ≠ 𝑏 → (((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) ∨ (𝑎 = 𝐵 ∧ 𝑏 = 𝐴)) → 𝐴 ≠ 𝐵))
108	107	ad2antrl 718	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})) → (((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) ∨ (𝑎 = 𝐵 ∧ 𝑏 = 𝐴)) → 𝐴 ≠ 𝐵))
109	97, 108	sylbid 232	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})) → ({𝑎, 𝑏} = {𝐴, 𝐵} → 𝐴 ≠ 𝐵))
110	109	adantr 474	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})) ∧ ((𝑎 = 𝐴 ∨ 𝑎 = 𝐵) ∧ (𝑏 = 𝐴 ∨ 𝑏 = 𝐵))) → ({𝑎, 𝑏} = {𝐴, 𝐵} → 𝐴 ≠ 𝐵))
111	92, 110	syl5bi 234	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})) ∧ ((𝑎 = 𝐴 ∨ 𝑎 = 𝐵) ∧ (𝑏 = 𝐴 ∨ 𝑏 = 𝐵))) → ((((𝐴 = 𝐴 ∨ 𝐴 = 𝐵) ∧ (𝐵 = 𝐴 ∨ 𝐵 = 𝐵)) → {𝑎, 𝑏} = {𝐴, 𝐵}) → 𝐴 ≠ 𝐵))
112	86, 111	sylbid 232	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})) ∧ ((𝑎 = 𝐴 ∨ 𝑎 = 𝐵) ∧ (𝑏 = 𝐴 ∨ 𝑏 = 𝐵))) → ((∀𝑥 ∈ {𝐴, 𝐵} (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → {𝑎, 𝑏} = {𝐴, 𝐵}) → 𝐴 ≠ 𝐵))
113	76, 112	syld 47	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})) ∧ ((𝑎 = 𝐴 ∨ 𝑎 = 𝐵) ∧ (𝑏 = 𝐴 ∨ 𝑏 = 𝐵))) → (∀𝑞 ∈ 𝑃 (∀𝑥 ∈ 𝑞 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → {𝑎, 𝑏} = 𝑞) → 𝐴 ≠ 𝐵))
114	113	ex 403	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})) → (((𝑎 = 𝐴 ∨ 𝑎 = 𝐵) ∧ (𝑏 = 𝐴 ∨ 𝑏 = 𝐵)) → (∀𝑞 ∈ 𝑃 (∀𝑥 ∈ 𝑞 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → {𝑎, 𝑏} = 𝑞) → 𝐴 ≠ 𝐵)))
115	114	impd 400	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})) → ((((𝑎 = 𝐴 ∨ 𝑎 = 𝐵) ∧ (𝑏 = 𝐴 ∨ 𝑏 = 𝐵)) ∧ ∀𝑞 ∈ 𝑃 (∀𝑥 ∈ 𝑞 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → {𝑎, 𝑏} = 𝑞)) → 𝐴 ≠ 𝐵))
116	22, 115	sylbid 232	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})) → ((∀𝑥 ∈ 𝑝 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ∧ ∀𝑞 ∈ 𝑃 (∀𝑥 ∈ 𝑞 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → 𝑝 = 𝑞)) → 𝐴 ≠ 𝐵))
117	116	ex 403	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ((𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏}) → ((∀𝑥 ∈ 𝑝 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ∧ ∀𝑞 ∈ 𝑃 (∀𝑥 ∈ 𝑞 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → 𝑝 = 𝑞)) → 𝐴 ≠ 𝐵)))
118	117	rexlimdvva 3220	. . . . . 6 ⊢ (𝜑 → (∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏}) → ((∀𝑥 ∈ 𝑝 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ∧ ∀𝑞 ∈ 𝑃 (∀𝑥 ∈ 𝑞 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → 𝑝 = 𝑞)) → 𝐴 ≠ 𝐵)))
119	6, 118	syl5bi 234	. . . . 5 ⊢ (𝜑 → (𝑝 ∈ 𝑃 → ((∀𝑥 ∈ 𝑝 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ∧ ∀𝑞 ∈ 𝑃 (∀𝑥 ∈ 𝑞 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → 𝑝 = 𝑞)) → 𝐴 ≠ 𝐵)))
120	119	imp 397	. . . 4 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑃) → ((∀𝑥 ∈ 𝑝 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ∧ ∀𝑞 ∈ 𝑃 (∀𝑥 ∈ 𝑞 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → 𝑝 = 𝑞)) → 𝐴 ≠ 𝐵))
121	120	rexlimdva 3212	. . 3 ⊢ (𝜑 → (∃𝑝 ∈ 𝑃 (∀𝑥 ∈ 𝑝 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ∧ ∀𝑞 ∈ 𝑃 (∀𝑥 ∈ 𝑞 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → 𝑝 = 𝑞)) → 𝐴 ≠ 𝐵))
122	2, 121	syl5bi 234	. 2 ⊢ (𝜑 → (∃!𝑝 ∈ 𝑃 ∀𝑥 ∈ 𝑝 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → 𝐴 ≠ 𝐵))
123	27	adantr 474	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → {𝐴, 𝐵} ∈ 𝒫 𝑉)
124		hashprg 13497	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴 ≠ 𝐵 ↔ (♯‘{𝐴, 𝐵}) = 2))
125	25, 124	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 ≠ 𝐵 ↔ (♯‘{𝐴, 𝐵}) = 2))
126	125	biimpd 221	. . . . . . . 8 ⊢ (𝜑 → (𝐴 ≠ 𝐵 → (♯‘{𝐴, 𝐵}) = 2))
127	126	imp 397	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → (♯‘{𝐴, 𝐵}) = 2)
128	123, 127	jca 507	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → ({𝐴, 𝐵} ∈ 𝒫 𝑉 ∧ (♯‘{𝐴, 𝐵}) = 2))
129	128, 70	sylibr 226	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → {𝐴, 𝐵} ∈ 𝑃)
130		raleq 3329	. . . . . . 7 ⊢ (𝑝 = {𝐴, 𝐵} → (∀𝑥 ∈ 𝑝 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ↔ ∀𝑥 ∈ {𝐴, 𝐵} (𝑥 = 𝐴 ∨ 𝑥 = 𝐵)))
131		eqeq1 2781	. . . . . . . . 9 ⊢ (𝑝 = {𝐴, 𝐵} → (𝑝 = 𝑦 ↔ {𝐴, 𝐵} = 𝑦))
132	131	imbi2d 332	. . . . . . . 8 ⊢ (𝑝 = {𝐴, 𝐵} → ((∀𝑥 ∈ 𝑦 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → 𝑝 = 𝑦) ↔ (∀𝑥 ∈ 𝑦 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → {𝐴, 𝐵} = 𝑦)))
133	132	ralbidv 3167	. . . . . . 7 ⊢ (𝑝 = {𝐴, 𝐵} → (∀𝑦 ∈ 𝑃 (∀𝑥 ∈ 𝑦 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → 𝑝 = 𝑦) ↔ ∀𝑦 ∈ 𝑃 (∀𝑥 ∈ 𝑦 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → {𝐴, 𝐵} = 𝑦)))
134	130, 133	anbi12d 624	. . . . . 6 ⊢ (𝑝 = {𝐴, 𝐵} → ((∀𝑥 ∈ 𝑝 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ∧ ∀𝑦 ∈ 𝑃 (∀𝑥 ∈ 𝑦 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → 𝑝 = 𝑦)) ↔ (∀𝑥 ∈ {𝐴, 𝐵} (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ∧ ∀𝑦 ∈ 𝑃 (∀𝑥 ∈ 𝑦 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → {𝐴, 𝐵} = 𝑦))))
135	134	adantl 475	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ 𝑝 = {𝐴, 𝐵}) → ((∀𝑥 ∈ 𝑝 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ∧ ∀𝑦 ∈ 𝑃 (∀𝑥 ∈ 𝑦 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → 𝑝 = 𝑦)) ↔ (∀𝑥 ∈ {𝐴, 𝐵} (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ∧ ∀𝑦 ∈ 𝑃 (∀𝑥 ∈ 𝑦 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → {𝐴, 𝐵} = 𝑦))))
136		vex 3400	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
137	136	elpr 4420	. . . . . . . . . 10 ⊢ (𝑥 ∈ {𝐴, 𝐵} ↔ (𝑥 = 𝐴 ∨ 𝑥 = 𝐵))
138	137	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → (𝑥 ∈ {𝐴, 𝐵} ↔ (𝑥 = 𝐴 ∨ 𝑥 = 𝐵)))
139	138	biimpd 221	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → (𝑥 ∈ {𝐴, 𝐵} → (𝑥 = 𝐴 ∨ 𝑥 = 𝐵)))
140	139	imp 397	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ 𝑥 ∈ {𝐴, 𝐵}) → (𝑥 = 𝐴 ∨ 𝑥 = 𝐵))
141	140	ralrimiva 3147	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → ∀𝑥 ∈ {𝐴, 𝐵} (𝑥 = 𝐴 ∨ 𝑥 = 𝐵))
142	3	eleq2i 2850	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝑃 ↔ 𝑦 ∈ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2})
143		elss2prb 13583	. . . . . . . . . 10 ⊢ (𝑦 ∈ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} ↔ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝑦 = {𝑎, 𝑏}))
144	142, 143	bitri 267	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝑃 ↔ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝑦 = {𝑎, 𝑏}))
145		prid1g 4526	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 ∈ 𝑉 → 𝑎 ∈ {𝑎, 𝑏})
146	145	ad2antrl 718	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → 𝑎 ∈ {𝑎, 𝑏})
147	146	adantr 474	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑦 = {𝑎, 𝑏})) → 𝑎 ∈ {𝑎, 𝑏})
148		eleq2 2847	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = {𝑎, 𝑏} → (𝑎 ∈ 𝑦 ↔ 𝑎 ∈ {𝑎, 𝑏}))
149	148	ad2antll 719	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑦 = {𝑎, 𝑏})) → (𝑎 ∈ 𝑦 ↔ 𝑎 ∈ {𝑎, 𝑏}))
150	147, 149	mpbird 249	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑦 = {𝑎, 𝑏})) → 𝑎 ∈ 𝑦)
151	12	rspcv 3506	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ 𝑦 → (∀𝑥 ∈ 𝑦 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → (𝑎 = 𝐴 ∨ 𝑎 = 𝐵)))
152	150, 151	syl 17	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑦 = {𝑎, 𝑏})) → (∀𝑥 ∈ 𝑦 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → (𝑎 = 𝐴 ∨ 𝑎 = 𝐵)))
153		prid2g 4527	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 ∈ 𝑉 → 𝑏 ∈ {𝑎, 𝑏})
154	153	ad2antll 719	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → 𝑏 ∈ {𝑎, 𝑏})
155	154	adantr 474	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑦 = {𝑎, 𝑏})) → 𝑏 ∈ {𝑎, 𝑏})
156		eleq2 2847	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = {𝑎, 𝑏} → (𝑏 ∈ 𝑦 ↔ 𝑏 ∈ {𝑎, 𝑏}))
157	156	ad2antll 719	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑦 = {𝑎, 𝑏})) → (𝑏 ∈ 𝑦 ↔ 𝑏 ∈ {𝑎, 𝑏}))
158	155, 157	mpbird 249	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑦 = {𝑎, 𝑏})) → 𝑏 ∈ 𝑦)
159	15	rspcv 3506	. . . . . . . . . . . . . 14 ⊢ (𝑏 ∈ 𝑦 → (∀𝑥 ∈ 𝑦 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → (𝑏 = 𝐴 ∨ 𝑏 = 𝐵)))
160	158, 159	syl 17	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑦 = {𝑎, 𝑏})) → (∀𝑥 ∈ 𝑦 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → (𝑏 = 𝐴 ∨ 𝑏 = 𝐵)))
161		simplrr 768	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑦 = {𝑎, 𝑏})) ∧ ((𝑏 = 𝐴 ∨ 𝑏 = 𝐵) ∧ (𝑎 = 𝐴 ∨ 𝑎 = 𝐵))) → 𝑦 = {𝑎, 𝑏})
162		eqtr3 2800	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐴) → 𝑎 = 𝑏)
163		eqneqall 2979	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑎 = 𝑏 → (𝑎 ≠ 𝑏 → {𝑎, 𝑏} = {𝐴, 𝐵}))
164	163	com12 32	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑎 ≠ 𝑏 → (𝑎 = 𝑏 → {𝑎, 𝑏} = {𝐴, 𝐵}))
165	164	ad2antrl 718	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑦 = {𝑎, 𝑏})) → (𝑎 = 𝑏 → {𝑎, 𝑏} = {𝐴, 𝐵}))
166	165	com12 32	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑎 = 𝑏 → ((((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑦 = {𝑎, 𝑏})) → {𝑎, 𝑏} = {𝐴, 𝐵}))
167	162, 166	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐴) → ((((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑦 = {𝑎, 𝑏})) → {𝑎, 𝑏} = {𝐴, 𝐵}))
168	167	ex 403	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = 𝐴 → (𝑏 = 𝐴 → ((((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑦 = {𝑎, 𝑏})) → {𝑎, 𝑏} = {𝐴, 𝐵})))
169	52	a1d 25	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑏 = 𝐴 ∧ 𝑎 = 𝐵) → ((((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑦 = {𝑎, 𝑏})) → {𝑎, 𝑏} = {𝐴, 𝐵}))
170	169	expcom 404	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = 𝐵 → (𝑏 = 𝐴 → ((((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑦 = {𝑎, 𝑏})) → {𝑎, 𝑏} = {𝐴, 𝐵})))
171	168, 170	jaoi 846	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑎 = 𝐴 ∨ 𝑎 = 𝐵) → (𝑏 = 𝐴 → ((((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑦 = {𝑎, 𝑏})) → {𝑎, 𝑏} = {𝐴, 𝐵})))
172	171	com12 32	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = 𝐴 → ((𝑎 = 𝐴 ∨ 𝑎 = 𝐵) → ((((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑦 = {𝑎, 𝑏})) → {𝑎, 𝑏} = {𝐴, 𝐵})))
173	44	a1d 25	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → ((((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑦 = {𝑎, 𝑏})) → {𝑎, 𝑏} = {𝐴, 𝐵}))
174	173	ex 403	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = 𝐴 → (𝑏 = 𝐵 → ((((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑦 = {𝑎, 𝑏})) → {𝑎, 𝑏} = {𝐴, 𝐵})))
175		eqtr3 2800	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑎 = 𝐵 ∧ 𝑏 = 𝐵) → 𝑎 = 𝑏)
176	175, 166	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑎 = 𝐵 ∧ 𝑏 = 𝐵) → ((((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑦 = {𝑎, 𝑏})) → {𝑎, 𝑏} = {𝐴, 𝐵}))
177	176	ex 403	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = 𝐵 → (𝑏 = 𝐵 → ((((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑦 = {𝑎, 𝑏})) → {𝑎, 𝑏} = {𝐴, 𝐵})))
178	174, 177	jaoi 846	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑎 = 𝐴 ∨ 𝑎 = 𝐵) → (𝑏 = 𝐵 → ((((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑦 = {𝑎, 𝑏})) → {𝑎, 𝑏} = {𝐴, 𝐵})))
179	178	com12 32	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = 𝐵 → ((𝑎 = 𝐴 ∨ 𝑎 = 𝐵) → ((((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑦 = {𝑎, 𝑏})) → {𝑎, 𝑏} = {𝐴, 𝐵})))
180	172, 179	jaoi 846	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑏 = 𝐴 ∨ 𝑏 = 𝐵) → ((𝑎 = 𝐴 ∨ 𝑎 = 𝐵) → ((((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑦 = {𝑎, 𝑏})) → {𝑎, 𝑏} = {𝐴, 𝐵})))
181	180	imp 397	. . . . . . . . . . . . . . . 16 ⊢ (((𝑏 = 𝐴 ∨ 𝑏 = 𝐵) ∧ (𝑎 = 𝐴 ∨ 𝑎 = 𝐵)) → ((((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑦 = {𝑎, 𝑏})) → {𝑎, 𝑏} = {𝐴, 𝐵}))
182	181	impcom 398	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑦 = {𝑎, 𝑏})) ∧ ((𝑏 = 𝐴 ∨ 𝑏 = 𝐵) ∧ (𝑎 = 𝐴 ∨ 𝑎 = 𝐵))) → {𝑎, 𝑏} = {𝐴, 𝐵})
183	161, 182	eqtr2d 2814	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑦 = {𝑎, 𝑏})) ∧ ((𝑏 = 𝐴 ∨ 𝑏 = 𝐵) ∧ (𝑎 = 𝐴 ∨ 𝑎 = 𝐵))) → {𝐴, 𝐵} = 𝑦)
184	183	exp32 413	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑦 = {𝑎, 𝑏})) → ((𝑏 = 𝐴 ∨ 𝑏 = 𝐵) → ((𝑎 = 𝐴 ∨ 𝑎 = 𝐵) → {𝐴, 𝐵} = 𝑦)))
185	160, 184	syld 47	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑦 = {𝑎, 𝑏})) → (∀𝑥 ∈ 𝑦 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → ((𝑎 = 𝐴 ∨ 𝑎 = 𝐵) → {𝐴, 𝐵} = 𝑦)))
186	152, 185	mpdd 43	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑦 = {𝑎, 𝑏})) → (∀𝑥 ∈ 𝑦 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → {𝐴, 𝐵} = 𝑦))
187	186	ex 403	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ((𝑎 ≠ 𝑏 ∧ 𝑦 = {𝑎, 𝑏}) → (∀𝑥 ∈ 𝑦 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → {𝐴, 𝐵} = 𝑦)))
188	187	rexlimdvva 3220	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → (∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝑦 = {𝑎, 𝑏}) → (∀𝑥 ∈ 𝑦 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → {𝐴, 𝐵} = 𝑦)))
189	144, 188	syl5bi 234	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → (𝑦 ∈ 𝑃 → (∀𝑥 ∈ 𝑦 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → {𝐴, 𝐵} = 𝑦)))
190	189	imp 397	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ 𝑦 ∈ 𝑃) → (∀𝑥 ∈ 𝑦 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → {𝐴, 𝐵} = 𝑦))
191	190	ralrimiva 3147	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → ∀𝑦 ∈ 𝑃 (∀𝑥 ∈ 𝑦 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → {𝐴, 𝐵} = 𝑦))
192	141, 191	jca 507	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → (∀𝑥 ∈ {𝐴, 𝐵} (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ∧ ∀𝑦 ∈ 𝑃 (∀𝑥 ∈ 𝑦 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → {𝐴, 𝐵} = 𝑦)))
193	129, 135, 192	rspcedvd 3517	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → ∃𝑝 ∈ 𝑃 (∀𝑥 ∈ 𝑝 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ∧ ∀𝑦 ∈ 𝑃 (∀𝑥 ∈ 𝑦 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → 𝑝 = 𝑦)))
194		raleq 3329	. . . . 5 ⊢ (𝑝 = 𝑦 → (∀𝑥 ∈ 𝑝 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ↔ ∀𝑥 ∈ 𝑦 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵)))
195	194	reu8 3613	. . . 4 ⊢ (∃!𝑝 ∈ 𝑃 ∀𝑥 ∈ 𝑝 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ↔ ∃𝑝 ∈ 𝑃 (∀𝑥 ∈ 𝑝 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ∧ ∀𝑦 ∈ 𝑃 (∀𝑥 ∈ 𝑦 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → 𝑝 = 𝑦)))
196	193, 195	sylibr 226	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → ∃!𝑝 ∈ 𝑃 ∀𝑥 ∈ 𝑝 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵))
197	196	ex 403	. 2 ⊢ (𝜑 → (𝐴 ≠ 𝐵 → ∃!𝑝 ∈ 𝑃 ∀𝑥 ∈ 𝑝 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵)))
198	122, 197	impbid 204	1 ⊢ (𝜑 → (∃!𝑝 ∈ 𝑃 ∀𝑥 ∈ 𝑝 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ↔ 𝐴 ≠ 𝐵))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   ∨ wo 836   = wceq 1601   ∈ wcel 2106   ≠ wne 2968  ∀wral 3089  ∃wrex 3090  ∃!wreu 3091  {crab 3093  Vcvv 3397  𝒫 cpw 4378  {cpr 4399  ‘cfv 6135  2c2 11430  ♯chash 13435

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-8 2108  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-cnex 10328  ax-resscn 10329  ax-1cn 10330  ax-icn 10331  ax-addcl 10332  ax-addrcl 10333  ax-mulcl 10334  ax-mulrcl 10335  ax-mulcom 10336  ax-addass 10337  ax-mulass 10338  ax-distr 10339  ax-i2m1 10340  ax-1ne0 10341  ax-1rid 10342  ax-rnegex 10343  ax-rrecex 10344  ax-cnre 10345  ax-pre-lttri 10346  ax-pre-lttrn 10347  ax-pre-ltadd 10348  ax-pre-mulgt0 10349

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-nel 3075  df-ral 3094  df-rex 3095  df-reu 3096  df-rmo 3097  df-rab 3098  df-v 3399  df-sbc 3652  df-csb 3751  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-pss 3807  df-nul 4141  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4672  df-int 4711  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-riota 6883  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-om 7344  df-1st 7445  df-2nd 7446  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-1o 7843  df-2o 7844  df-oadd 7847  df-er 8026  df-en 8242  df-dom 8243  df-sdom 8244  df-fin 8245  df-card 9098  df-cda 9325  df-pnf 10413  df-mnf 10414  df-xr 10415  df-ltxr 10416  df-le 10417  df-sub 10608  df-neg 10609  df-nn 11375  df-2 11438  df-n0 11643  df-z 11729  df-uz 11993  df-fz 12644  df-hash 13436

This theorem is referenced by:  requad2  42553