Theorem paireqne 46165

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 3322	. . . 4 ⊢ (𝑝 = 𝑞 → (∀𝑥 ∈ 𝑝 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ↔ ∀𝑥 ∈ 𝑞 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵)))
2	1	reu8 3728	. . 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 2825	. . . . . . 7 ⊢ (𝑝 ∈ 𝑃 ↔ 𝑝 ∈ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2})
5		elss2prb 14444	. . . . . . 7 ⊢ (𝑝 ∈ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} ↔ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏}))
6	4, 5	bitri 274	. . . . . 6 ⊢ (𝑝 ∈ 𝑃 ↔ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏}))
7		raleq 3322	. . . . . . . . . . . 12 ⊢ (𝑝 = {𝑎, 𝑏} → (∀𝑥 ∈ 𝑝 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ↔ ∀𝑥 ∈ {𝑎, 𝑏} (𝑥 = 𝐴 ∨ 𝑥 = 𝐵)))
8		vex 3478	. . . . . . . . . . . . 13 ⊢ 𝑎 ∈ V
9		vex 3478	. . . . . . . . . . . . 13 ⊢ 𝑏 ∈ V
10		eqeq1 2736	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑎 → (𝑥 = 𝐴 ↔ 𝑎 = 𝐴))
11		eqeq1 2736	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑎 → (𝑥 = 𝐵 ↔ 𝑎 = 𝐵))
12	10, 11	orbi12d 917	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑎 → ((𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ↔ (𝑎 = 𝐴 ∨ 𝑎 = 𝐵)))
13		eqeq1 2736	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑏 → (𝑥 = 𝐴 ↔ 𝑏 = 𝐴))
14		eqeq1 2736	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑏 → (𝑥 = 𝐵 ↔ 𝑏 = 𝐵))
15	13, 14	orbi12d 917	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑏 → ((𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ↔ (𝑏 = 𝐴 ∨ 𝑏 = 𝐵)))
16	8, 9, 12, 15	ralpr 4703	. . . . . . . . . . . 12 ⊢ (∀𝑥 ∈ {𝑎, 𝑏} (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ↔ ((𝑎 = 𝐴 ∨ 𝑎 = 𝐵) ∧ (𝑏 = 𝐴 ∨ 𝑏 = 𝐵)))
17	7, 16	bitrdi 286	. . . . . . . . . . 11 ⊢ (𝑝 = {𝑎, 𝑏} → (∀𝑥 ∈ 𝑝 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ↔ ((𝑎 = 𝐴 ∨ 𝑎 = 𝐵) ∧ (𝑏 = 𝐴 ∨ 𝑏 = 𝐵))))
18		eqeq1 2736	. . . . . . . . . . . . 13 ⊢ (𝑝 = {𝑎, 𝑏} → (𝑝 = 𝑞 ↔ {𝑎, 𝑏} = 𝑞))
19	18	imbi2d 340	. . . . . . . . . . . 12 ⊢ (𝑝 = {𝑎, 𝑏} → ((∀𝑥 ∈ 𝑞 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → 𝑝 = 𝑞) ↔ (∀𝑥 ∈ 𝑞 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → {𝑎, 𝑏} = 𝑞)))
20	19	ralbidv 3177	. . . . . . . . . . 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 631	. . . . . . . . . 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 727	. . . . . . . . 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 512	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉))
26		prelpwi 5446	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → {𝐴, 𝐵} ∈ 𝒫 𝑉)
27	25, 26	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → {𝐴, 𝐵} ∈ 𝒫 𝑉)
28	27	ad3antrrr 728	. . . . . . . . . . . . . . 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 14351	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → (𝑎 ≠ 𝑏 ↔ (♯‘{𝑎, 𝑏}) = 2))
30	29	adantl 482	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → (𝑎 ≠ 𝑏 ↔ (♯‘{𝑎, 𝑏}) = 2))
31	30	biimpd 228	. . . . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏}) → ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → (♯‘{𝑎, 𝑏}) = 2))
34	33	impcom 408	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})) → (♯‘{𝑎, 𝑏}) = 2)
35	34	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})) ∧ ((𝑎 = 𝐴 ∨ 𝑎 = 𝐵) ∧ (𝑏 = 𝐴 ∨ 𝑏 = 𝐵))) → (♯‘{𝑎, 𝑏}) = 2)
36		eqtr3 2758	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑏 = 𝐴 ∧ 𝑎 = 𝐴) → 𝑏 = 𝑎)
37		eqneqall 2951	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑎 = 𝑏 → (𝑎 ≠ 𝑏 → (𝑝 = {𝑎, 𝑏} → {𝐴, 𝐵} = {𝑎, 𝑏})))
38	37	impd 411	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 411	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑎 = 𝑏 → (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})) → {𝐴, 𝐵} = {𝑎, 𝑏}))
41	40	equcoms 2023	. . . . . . . . . . . . . . . . . . . . . . . 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 413	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑏 = 𝐴 → (𝑎 = 𝐴 → (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})) → {𝐴, 𝐵} = {𝑎, 𝑏})))
44		preq12 4738	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → {𝑎, 𝑏} = {𝐴, 𝐵})
45	44	eqcomd 2738	. . . . . . . . . . . . . . . . . . . . . . . 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 414	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑏 = 𝐵 → (𝑎 = 𝐴 → (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})) → {𝐴, 𝐵} = {𝑎, 𝑏})))
48	43, 47	jaoi 855	. . . . . . . . . . . . . . . . . . . . 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 4735	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ {𝑎, 𝑏} = {𝑏, 𝑎}
51		preq12 4738	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑏 = 𝐴 ∧ 𝑎 = 𝐵) → {𝑏, 𝑎} = {𝐴, 𝐵})
52	50, 51	eqtrid 2784	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑏 = 𝐴 ∧ 𝑎 = 𝐵) → {𝑎, 𝑏} = {𝐴, 𝐵})
53	52	eqcomd 2738	. . . . . . . . . . . . . . . . . . . . . . . 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 413	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑏 = 𝐴 → (𝑎 = 𝐵 → (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})) → {𝐴, 𝐵} = {𝑎, 𝑏})))
56		eqtr3 2758	. . . . . . . . . . . . . . . . . . . . . . . 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 413	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑏 = 𝐵 → (𝑎 = 𝐵 → (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})) → {𝐴, 𝐵} = {𝑎, 𝑏})))
59	55, 58	jaoi 855	. . . . . . . . . . . . . . . . . . . . 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 855	. . . . . . . . . . . . . . . . . . 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 407	. . . . . . . . . . . . . . . . . 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 408	. . . . . . . . . . . . . . . . 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 6896	. . . . . . . . . . . . . . . 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 256	. . . . . . . . . . . . . . 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 512	. . . . . . . . . . . . . 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 2825	. . . . . . . . . . . . . . 15 ⊢ ({𝐴, 𝐵} ∈ 𝑃 ↔ {𝐴, 𝐵} ∈ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2})
68		fveqeq2 6897	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = {𝐴, 𝐵} → ((♯‘𝑥) = 2 ↔ (♯‘{𝐴, 𝐵}) = 2))
69	68	elrab 3682	. . . . . . . . . . . . . . 15 ⊢ ({𝐴, 𝐵} ∈ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} ↔ ({𝐴, 𝐵} ∈ 𝒫 𝑉 ∧ (♯‘{𝐴, 𝐵}) = 2))
70	67, 69	bitri 274	. . . . . . . . . . . . . 14 ⊢ ({𝐴, 𝐵} ∈ 𝑃 ↔ ({𝐴, 𝐵} ∈ 𝒫 𝑉 ∧ (♯‘{𝐴, 𝐵}) = 2))
71	66, 70	sylibr 233	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})) ∧ ((𝑎 = 𝐴 ∨ 𝑎 = 𝐵) ∧ (𝑏 = 𝐴 ∨ 𝑏 = 𝐵))) → {𝐴, 𝐵} ∈ 𝑃)
72		raleq 3322	. . . . . . . . . . . . . . 15 ⊢ (𝑞 = {𝐴, 𝐵} → (∀𝑥 ∈ 𝑞 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ↔ ∀𝑥 ∈ {𝐴, 𝐵} (𝑥 = 𝐴 ∨ 𝑥 = 𝐵)))
73		eqeq2 2744	. . . . . . . . . . . . . . 15 ⊢ (𝑞 = {𝐴, 𝐵} → ({𝑎, 𝑏} = 𝑞 ↔ {𝑎, 𝑏} = {𝐴, 𝐵}))
74	72, 73	imbi12d 344	. . . . . . . . . . . . . 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 3608	. . . . . . . . . . . . 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 2736	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝐴 ↔ 𝐴 = 𝐴))
78		eqeq1 2736	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝐵 ↔ 𝐴 = 𝐵))
79	77, 78	orbi12d 917	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝐴 → ((𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ↔ (𝐴 = 𝐴 ∨ 𝐴 = 𝐵)))
80		eqeq1 2736	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝐵 → (𝑥 = 𝐴 ↔ 𝐵 = 𝐴))
81		eqeq1 2736	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝐵 → (𝑥 = 𝐵 ↔ 𝐵 = 𝐵))
82	80, 81	orbi12d 917	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝐵 → ((𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ↔ (𝐵 = 𝐴 ∨ 𝐵 = 𝐵)))
83	79, 82	ralprg 4697	. . . . . . . . . . . . . . . 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 341	. . . . . . . . . . . . . 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 728	. . . . . . . . . . . . 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 2732	. . . . . . . . . . . . . . . 16 ⊢ 𝐴 = 𝐴
88	87	orci 863	. . . . . . . . . . . . . . 15 ⊢ (𝐴 = 𝐴 ∨ 𝐴 = 𝐵)
89		eqid 2732	. . . . . . . . . . . . . . . 16 ⊢ 𝐵 = 𝐵
90	89	olci 864	. . . . . . . . . . . . . . 15 ⊢ (𝐵 = 𝐴 ∨ 𝐵 = 𝐵)
91		pm5.5 361	. . . . . . . . . . . . . . 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 690	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 = 𝐴 ∨ 𝐴 = 𝐵) ∧ (𝐵 = 𝐴 ∨ 𝐵 = 𝐵)) → {𝑎, 𝑏} = {𝐴, 𝐵}) ↔ {𝑎, 𝑏} = {𝐴, 𝐵})
93	8, 9	pm3.2i 471	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 ∈ V ∧ 𝑏 ∈ V)
94		preq12bg 4853	. . . . . . . . . . . . . . . . . . 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 587	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ({𝑎, 𝑏} = {𝐴, 𝐵} ↔ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) ∨ (𝑎 = 𝐵 ∧ 𝑏 = 𝐴))))
96	95	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ({𝑎, 𝑏} = {𝐴, 𝐵} ↔ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) ∨ (𝑎 = 𝐵 ∧ 𝑏 = 𝐴))))
97	96	adantr 481	. . . . . . . . . . . . . . . 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 2749	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (𝑎 = 𝑏 ↔ 𝐴 = 𝐵))
99	98	necon3bid 2985	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (𝑎 ≠ 𝑏 ↔ 𝐴 ≠ 𝐵))
100	99	biimpd 228	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (𝑎 ≠ 𝑏 → 𝐴 ≠ 𝐵))
101		eqeq12 2749	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑎 = 𝐵 ∧ 𝑏 = 𝐴) → (𝑎 = 𝑏 ↔ 𝐵 = 𝐴))
102	101	necon3bid 2985	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑎 = 𝐵 ∧ 𝑏 = 𝐴) → (𝑎 ≠ 𝑏 ↔ 𝐵 ≠ 𝐴))
103	102	biimpd 228	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑎 = 𝐵 ∧ 𝑏 = 𝐴) → (𝑎 ≠ 𝑏 → 𝐵 ≠ 𝐴))
104		necom 2994	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐴 ≠ 𝐵 ↔ 𝐵 ≠ 𝐴)
105	103, 104	syl6ibr 251	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑎 = 𝐵 ∧ 𝑏 = 𝐴) → (𝑎 ≠ 𝑏 → 𝐴 ≠ 𝐵))
106	100, 105	jaoi 855	. . . . . . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . . 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 239	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})) → ({𝑎, 𝑏} = {𝐴, 𝐵} → 𝐴 ≠ 𝐵))
110	109	adantr 481	. . . . . . . . . . . . . 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	biimtrid 241	. . . . . . . . . . . . 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 239	. . . . . . . . . . . 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 413	. . . . . . . . . 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 411	. . . . . . . . 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 239	. . . . . . . 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 413	. . . . . . 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 3211	. . . . . 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	biimtrid 241	. . . . 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 407	. . . 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 3155	. . 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	biimtrid 241	. 2 ⊢ (𝜑 → (∃!𝑝 ∈ 𝑃 ∀𝑥 ∈ 𝑝 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → 𝐴 ≠ 𝐵))
123	27	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → {𝐴, 𝐵} ∈ 𝒫 𝑉)
124		hashprg 14351	. . . . . . . . . 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 228	. . . . . . . 8 ⊢ (𝜑 → (𝐴 ≠ 𝐵 → (♯‘{𝐴, 𝐵}) = 2))
127	126	imp 407	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → (♯‘{𝐴, 𝐵}) = 2)
128	123, 127	jca 512	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → ({𝐴, 𝐵} ∈ 𝒫 𝑉 ∧ (♯‘{𝐴, 𝐵}) = 2))
129	128, 70	sylibr 233	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → {𝐴, 𝐵} ∈ 𝑃)
130		raleq 3322	. . . . . . 7 ⊢ (𝑝 = {𝐴, 𝐵} → (∀𝑥 ∈ 𝑝 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ↔ ∀𝑥 ∈ {𝐴, 𝐵} (𝑥 = 𝐴 ∨ 𝑥 = 𝐵)))
131		eqeq1 2736	. . . . . . . . 9 ⊢ (𝑝 = {𝐴, 𝐵} → (𝑝 = 𝑦 ↔ {𝐴, 𝐵} = 𝑦))
132	131	imbi2d 340	. . . . . . . 8 ⊢ (𝑝 = {𝐴, 𝐵} → ((∀𝑥 ∈ 𝑦 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → 𝑝 = 𝑦) ↔ (∀𝑥 ∈ 𝑦 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → {𝐴, 𝐵} = 𝑦)))
133	132	ralbidv 3177	. . . . . . 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 631	. . . . . 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 482	. . . . 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 3478	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
137	136	elpr 4650	. . . . . . . . . 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 228	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → (𝑥 ∈ {𝐴, 𝐵} → (𝑥 = 𝐴 ∨ 𝑥 = 𝐵)))
140	139	imp 407	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ 𝑥 ∈ {𝐴, 𝐵}) → (𝑥 = 𝐴 ∨ 𝑥 = 𝐵))
141	140	ralrimiva 3146	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → ∀𝑥 ∈ {𝐴, 𝐵} (𝑥 = 𝐴 ∨ 𝑥 = 𝐵))
142	3	eleq2i 2825	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝑃 ↔ 𝑦 ∈ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2})
143		elss2prb 14444	. . . . . . . . . 10 ⊢ (𝑦 ∈ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} ↔ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝑦 = {𝑎, 𝑏}))
144	142, 143	bitri 274	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝑃 ↔ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝑦 = {𝑎, 𝑏}))
145		prid1g 4763	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 ∈ 𝑉 → 𝑎 ∈ {𝑎, 𝑏})
146	145	ad2antrl 726	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → 𝑎 ∈ {𝑎, 𝑏})
147	146	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑦 = {𝑎, 𝑏})) → 𝑎 ∈ {𝑎, 𝑏})
148		eleq2 2822	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = {𝑎, 𝑏} → (𝑎 ∈ 𝑦 ↔ 𝑎 ∈ {𝑎, 𝑏}))
149	148	ad2antll 727	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑦 = {𝑎, 𝑏})) → (𝑎 ∈ 𝑦 ↔ 𝑎 ∈ {𝑎, 𝑏}))
150	147, 149	mpbird 256	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑦 = {𝑎, 𝑏})) → 𝑎 ∈ 𝑦)
151	12	rspcv 3608	. . . . . . . . . . . . 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 4764	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 ∈ 𝑉 → 𝑏 ∈ {𝑎, 𝑏})
154	153	ad2antll 727	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → 𝑏 ∈ {𝑎, 𝑏})
155	154	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑦 = {𝑎, 𝑏})) → 𝑏 ∈ {𝑎, 𝑏})
156		eleq2 2822	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = {𝑎, 𝑏} → (𝑏 ∈ 𝑦 ↔ 𝑏 ∈ {𝑎, 𝑏}))
157	156	ad2antll 727	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑦 = {𝑎, 𝑏})) → (𝑏 ∈ 𝑦 ↔ 𝑏 ∈ {𝑎, 𝑏}))
158	155, 157	mpbird 256	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑦 = {𝑎, 𝑏})) → 𝑏 ∈ 𝑦)
159	15	rspcv 3608	. . . . . . . . . . . . . 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 776	. . . . . . . . . . . . . . 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 2758	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐴) → 𝑎 = 𝑏)
163		eqneqall 2951	. . . . . . . . . . . . . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . . . . . . . . . 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 413	. . . . . . . . . . . . . . . . . . . 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 414	. . . . . . . . . . . . . . . . . . . 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 855	. . . . . . . . . . . . . . . . . . 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 413	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = 𝐴 → (𝑏 = 𝐵 → ((((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑦 = {𝑎, 𝑏})) → {𝑎, 𝑏} = {𝐴, 𝐵})))
175		eqtr3 2758	. . . . . . . . . . . . . . . . . . . . . 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 413	. . . . . . . . . . . . . . . . . . . 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 855	. . . . . . . . . . . . . . . . . . 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 855	. . . . . . . . . . . . . . . . 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 407	. . . . . . . . . . . . . . . 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 408	. . . . . . . . . . . . . . 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 2773	. . . . . . . . . . . . . 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 421	. . . . . . . . . . . . 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 413	. . . . . . . . . 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 3211	. . . . . . . . 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	biimtrid 241	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → (𝑦 ∈ 𝑃 → (∀𝑥 ∈ 𝑦 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → {𝐴, 𝐵} = 𝑦)))
190	189	imp 407	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ 𝑦 ∈ 𝑃) → (∀𝑥 ∈ 𝑦 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → {𝐴, 𝐵} = 𝑦))
191	190	ralrimiva 3146	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → ∀𝑦 ∈ 𝑃 (∀𝑥 ∈ 𝑦 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → {𝐴, 𝐵} = 𝑦))
192	141, 191	jca 512	. . . . 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 3614	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → ∃𝑝 ∈ 𝑃 (∀𝑥 ∈ 𝑝 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ∧ ∀𝑦 ∈ 𝑃 (∀𝑥 ∈ 𝑦 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → 𝑝 = 𝑦)))
194		raleq 3322	. . . . 5 ⊢ (𝑝 = 𝑦 → (∀𝑥 ∈ 𝑝 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ↔ ∀𝑥 ∈ 𝑦 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵)))
195	194	reu8 3728	. . . 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 233	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → ∃!𝑝 ∈ 𝑃 ∀𝑥 ∈ 𝑝 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵))
197	196	ex 413	. 2 ⊢ (𝜑 → (𝐴 ≠ 𝐵 → ∃!𝑝 ∈ 𝑃 ∀𝑥 ∈ 𝑝 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵)))
198	122, 197	impbid 211	1 ⊢ (𝜑 → (∃!𝑝 ∈ 𝑃 ∀𝑥 ∈ 𝑝 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ↔ 𝐴 ≠ 𝐵))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 845   = wceq 1541   ∈ wcel 2106   ≠ wne 2940  ∀wral 3061  ∃wrex 3070  ∃!wreu 3374  {crab 3432  Vcvv 3474  𝒫 cpw 4601  {cpr 4629  ‘cfv 6540  2c2 12263  ♯chash 14286

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-2o 8463  df-oadd 8466  df-er 8699  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-dju 9892  df-card 9930  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-nn 12209  df-2 12271  df-n0 12469  df-z 12555  df-uz 12819  df-fz 13481  df-hash 14287

This theorem is referenced by:  requad2  46277