Theorem paireqne 47505

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 3286	. . . 4 ⊢ (𝑝 = 𝑞 → (∀𝑥 ∈ 𝑝 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ↔ ∀𝑥 ∈ 𝑞 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵)))
2	1	reu8 3693	. . 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 2820	. . . . . . 7 ⊢ (𝑝 ∈ 𝑃 ↔ 𝑝 ∈ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2})
5		elss2prb 14395	. . . . . . 7 ⊢ (𝑝 ∈ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} ↔ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏}))
6	4, 5	bitri 275	. . . . . 6 ⊢ (𝑝 ∈ 𝑃 ↔ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏}))
7		raleq 3286	. . . . . . . . . . . 12 ⊢ (𝑝 = {𝑎, 𝑏} → (∀𝑥 ∈ 𝑝 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ↔ ∀𝑥 ∈ {𝑎, 𝑏} (𝑥 = 𝐴 ∨ 𝑥 = 𝐵)))
8		vex 3440	. . . . . . . . . . . . 13 ⊢ 𝑎 ∈ V
9		vex 3440	. . . . . . . . . . . . 13 ⊢ 𝑏 ∈ V
10		eqeq1 2733	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑎 → (𝑥 = 𝐴 ↔ 𝑎 = 𝐴))
11		eqeq1 2733	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑎 → (𝑥 = 𝐵 ↔ 𝑎 = 𝐵))
12	10, 11	orbi12d 918	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑎 → ((𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ↔ (𝑎 = 𝐴 ∨ 𝑎 = 𝐵)))
13		eqeq1 2733	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑏 → (𝑥 = 𝐴 ↔ 𝑏 = 𝐴))
14		eqeq1 2733	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑏 → (𝑥 = 𝐵 ↔ 𝑏 = 𝐵))
15	13, 14	orbi12d 918	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑏 → ((𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ↔ (𝑏 = 𝐴 ∨ 𝑏 = 𝐵)))
16	8, 9, 12, 15	ralpr 4652	. . . . . . . . . . . 12 ⊢ (∀𝑥 ∈ {𝑎, 𝑏} (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ↔ ((𝑎 = 𝐴 ∨ 𝑎 = 𝐵) ∧ (𝑏 = 𝐴 ∨ 𝑏 = 𝐵)))
17	7, 16	bitrdi 287	. . . . . . . . . . 11 ⊢ (𝑝 = {𝑎, 𝑏} → (∀𝑥 ∈ 𝑝 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ↔ ((𝑎 = 𝐴 ∨ 𝑎 = 𝐵) ∧ (𝑏 = 𝐴 ∨ 𝑏 = 𝐵))))
18		eqeq1 2733	. . . . . . . . . . . . 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 3152	. . . . . . . . . . 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 632	. . . . . . . . . 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 729	. . . . . . . . 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 511	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉))
26		prelpwi 5390	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → {𝐴, 𝐵} ∈ 𝒫 𝑉)
27	25, 26	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → {𝐴, 𝐵} ∈ 𝒫 𝑉)
28	27	ad3antrrr 730	. . . . . . . . . . . . . . 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 14302	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → (𝑎 ≠ 𝑏 ↔ (♯‘{𝑎, 𝑏}) = 2))
30	29	adantl 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → (𝑎 ≠ 𝑏 ↔ (♯‘{𝑎, 𝑏}) = 2))
31	30	biimpd 229	. . . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏}) → ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → (♯‘{𝑎, 𝑏}) = 2))
34	33	impcom 407	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})) → (♯‘{𝑎, 𝑏}) = 2)
35	34	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})) ∧ ((𝑎 = 𝐴 ∨ 𝑎 = 𝐵) ∧ (𝑏 = 𝐴 ∨ 𝑏 = 𝐵))) → (♯‘{𝑎, 𝑏}) = 2)
36		eqtr3 2751	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑏 = 𝐴 ∧ 𝑎 = 𝐴) → 𝑏 = 𝑎)
37		eqneqall 2936	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑎 = 𝑏 → (𝑎 ≠ 𝑏 → (𝑝 = {𝑎, 𝑏} → {𝐴, 𝐵} = {𝑎, 𝑏})))
38	37	impd 410	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 410	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑎 = 𝑏 → (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})) → {𝐴, 𝐵} = {𝑎, 𝑏}))
41	40	equcoms 2020	. . . . . . . . . . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑏 = 𝐴 → (𝑎 = 𝐴 → (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})) → {𝐴, 𝐵} = {𝑎, 𝑏})))
44		preq12 4687	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → {𝑎, 𝑏} = {𝐴, 𝐵})
45	44	eqcomd 2735	. . . . . . . . . . . . . . . . . . . . . . . 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 413	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑏 = 𝐵 → (𝑎 = 𝐴 → (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})) → {𝐴, 𝐵} = {𝑎, 𝑏})))
48	43, 47	jaoi 857	. . . . . . . . . . . . . . . . . . . . 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 4684	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ {𝑎, 𝑏} = {𝑏, 𝑎}
51		preq12 4687	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑏 = 𝐴 ∧ 𝑎 = 𝐵) → {𝑏, 𝑎} = {𝐴, 𝐵})
52	50, 51	eqtrid 2776	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑏 = 𝐴 ∧ 𝑎 = 𝐵) → {𝑎, 𝑏} = {𝐴, 𝐵})
53	52	eqcomd 2735	. . . . . . . . . . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑏 = 𝐴 → (𝑎 = 𝐵 → (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})) → {𝐴, 𝐵} = {𝑎, 𝑏})))
56		eqtr3 2751	. . . . . . . . . . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑏 = 𝐵 → (𝑎 = 𝐵 → (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})) → {𝐴, 𝐵} = {𝑎, 𝑏})))
59	55, 58	jaoi 857	. . . . . . . . . . . . . . . . . . . . 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 857	. . . . . . . . . . . . . . . . . . 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 406	. . . . . . . . . . . . . . . . . 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 407	. . . . . . . . . . . . . . . . 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 6830	. . . . . . . . . . . . . . . 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 257	. . . . . . . . . . . . . . 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 511	. . . . . . . . . . . . . 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 2820	. . . . . . . . . . . . . . 15 ⊢ ({𝐴, 𝐵} ∈ 𝑃 ↔ {𝐴, 𝐵} ∈ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2})
68		fveqeq2 6831	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = {𝐴, 𝐵} → ((♯‘𝑥) = 2 ↔ (♯‘{𝐴, 𝐵}) = 2))
69	68	elrab 3648	. . . . . . . . . . . . . . 15 ⊢ ({𝐴, 𝐵} ∈ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} ↔ ({𝐴, 𝐵} ∈ 𝒫 𝑉 ∧ (♯‘{𝐴, 𝐵}) = 2))
70	67, 69	bitri 275	. . . . . . . . . . . . . 14 ⊢ ({𝐴, 𝐵} ∈ 𝑃 ↔ ({𝐴, 𝐵} ∈ 𝒫 𝑉 ∧ (♯‘{𝐴, 𝐵}) = 2))
71	66, 70	sylibr 234	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})) ∧ ((𝑎 = 𝐴 ∨ 𝑎 = 𝐵) ∧ (𝑏 = 𝐴 ∨ 𝑏 = 𝐵))) → {𝐴, 𝐵} ∈ 𝑃)
72		raleq 3286	. . . . . . . . . . . . . . 15 ⊢ (𝑞 = {𝐴, 𝐵} → (∀𝑥 ∈ 𝑞 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ↔ ∀𝑥 ∈ {𝐴, 𝐵} (𝑥 = 𝐴 ∨ 𝑥 = 𝐵)))
73		eqeq2 2741	. . . . . . . . . . . . . . 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 3573	. . . . . . . . . . . . 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 2733	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝐴 ↔ 𝐴 = 𝐴))
78		eqeq1 2733	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝐵 ↔ 𝐴 = 𝐵))
79	77, 78	orbi12d 918	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝐴 → ((𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ↔ (𝐴 = 𝐴 ∨ 𝐴 = 𝐵)))
80		eqeq1 2733	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝐵 → (𝑥 = 𝐴 ↔ 𝐵 = 𝐴))
81		eqeq1 2733	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝐵 → (𝑥 = 𝐵 ↔ 𝐵 = 𝐵))
82	80, 81	orbi12d 918	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝐵 → ((𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ↔ (𝐵 = 𝐴 ∨ 𝐵 = 𝐵)))
83	79, 82	ralprg 4648	. . . . . . . . . . . . . . . 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 730	. . . . . . . . . . . . 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 2729	. . . . . . . . . . . . . . . 16 ⊢ 𝐴 = 𝐴
88	87	orci 865	. . . . . . . . . . . . . . 15 ⊢ (𝐴 = 𝐴 ∨ 𝐴 = 𝐵)
89		eqid 2729	. . . . . . . . . . . . . . . 16 ⊢ 𝐵 = 𝐵
90	89	olci 866	. . . . . . . . . . . . . . 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 692	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 = 𝐴 ∨ 𝐴 = 𝐵) ∧ (𝐵 = 𝐴 ∨ 𝐵 = 𝐵)) → {𝑎, 𝑏} = {𝐴, 𝐵}) ↔ {𝑎, 𝑏} = {𝐴, 𝐵})
93	8, 9	pm3.2i 470	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 ∈ V ∧ 𝑏 ∈ V)
94		preq12bg 4804	. . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ({𝑎, 𝑏} = {𝐴, 𝐵} ↔ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) ∨ (𝑎 = 𝐵 ∧ 𝑏 = 𝐴))))
97	96	adantr 480	. . . . . . . . . . . . . . . 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 2746	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (𝑎 = 𝑏 ↔ 𝐴 = 𝐵))
99	98	necon3bid 2969	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (𝑎 ≠ 𝑏 ↔ 𝐴 ≠ 𝐵))
100	99	biimpd 229	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (𝑎 ≠ 𝑏 → 𝐴 ≠ 𝐵))
101		eqeq12 2746	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑎 = 𝐵 ∧ 𝑏 = 𝐴) → (𝑎 = 𝑏 ↔ 𝐵 = 𝐴))
102	101	necon3bid 2969	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑎 = 𝐵 ∧ 𝑏 = 𝐴) → (𝑎 ≠ 𝑏 ↔ 𝐵 ≠ 𝐴))
103	102	biimpd 229	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑎 = 𝐵 ∧ 𝑏 = 𝐴) → (𝑎 ≠ 𝑏 → 𝐵 ≠ 𝐴))
104		necom 2978	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐴 ≠ 𝐵 ↔ 𝐵 ≠ 𝐴)
105	103, 104	imbitrrdi 252	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑎 = 𝐵 ∧ 𝑏 = 𝐴) → (𝑎 ≠ 𝑏 → 𝐴 ≠ 𝐵))
106	100, 105	jaoi 857	. . . . . . . . . . . . . . . . . 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 728	. . . . . . . . . . . . . . . 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 240	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})) → ({𝑎, 𝑏} = {𝐴, 𝐵} → 𝐴 ≠ 𝐵))
110	109	adantr 480	. . . . . . . . . . . . . 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 242	. . . . . . . . . . . . 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 240	. . . . . . . . . . . 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 412	. . . . . . . . . 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 410	. . . . . . . . 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 240	. . . . . . . 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 412	. . . . . . 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 3186	. . . . . 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 242	. . . . 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 406	. . . 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 3130	. . 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 242	. 2 ⊢ (𝜑 → (∃!𝑝 ∈ 𝑃 ∀𝑥 ∈ 𝑝 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → 𝐴 ≠ 𝐵))
123	27	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → {𝐴, 𝐵} ∈ 𝒫 𝑉)
124		hashprg 14302	. . . . . . . . . 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 229	. . . . . . . 8 ⊢ (𝜑 → (𝐴 ≠ 𝐵 → (♯‘{𝐴, 𝐵}) = 2))
127	126	imp 406	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → (♯‘{𝐴, 𝐵}) = 2)
128	123, 127	jca 511	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → ({𝐴, 𝐵} ∈ 𝒫 𝑉 ∧ (♯‘{𝐴, 𝐵}) = 2))
129	128, 70	sylibr 234	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → {𝐴, 𝐵} ∈ 𝑃)
130		raleq 3286	. . . . . . 7 ⊢ (𝑝 = {𝐴, 𝐵} → (∀𝑥 ∈ 𝑝 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ↔ ∀𝑥 ∈ {𝐴, 𝐵} (𝑥 = 𝐴 ∨ 𝑥 = 𝐵)))
131		eqeq1 2733	. . . . . . . . 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 3152	. . . . . . 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 632	. . . . . 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 481	. . . . 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 3440	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
137	136	elpr 4602	. . . . . . . . . 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 229	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → (𝑥 ∈ {𝐴, 𝐵} → (𝑥 = 𝐴 ∨ 𝑥 = 𝐵)))
140	139	imp 406	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ 𝑥 ∈ {𝐴, 𝐵}) → (𝑥 = 𝐴 ∨ 𝑥 = 𝐵))
141	140	ralrimiva 3121	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → ∀𝑥 ∈ {𝐴, 𝐵} (𝑥 = 𝐴 ∨ 𝑥 = 𝐵))
142	3	eleq2i 2820	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝑃 ↔ 𝑦 ∈ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2})
143		elss2prb 14395	. . . . . . . . . 10 ⊢ (𝑦 ∈ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} ↔ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝑦 = {𝑎, 𝑏}))
144	142, 143	bitri 275	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝑃 ↔ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝑦 = {𝑎, 𝑏}))
145		prid1g 4712	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 ∈ 𝑉 → 𝑎 ∈ {𝑎, 𝑏})
146	145	ad2antrl 728	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → 𝑎 ∈ {𝑎, 𝑏})
147	146	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑦 = {𝑎, 𝑏})) → 𝑎 ∈ {𝑎, 𝑏})
148		eleq2 2817	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = {𝑎, 𝑏} → (𝑎 ∈ 𝑦 ↔ 𝑎 ∈ {𝑎, 𝑏}))
149	148	ad2antll 729	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑦 = {𝑎, 𝑏})) → (𝑎 ∈ 𝑦 ↔ 𝑎 ∈ {𝑎, 𝑏}))
150	147, 149	mpbird 257	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑦 = {𝑎, 𝑏})) → 𝑎 ∈ 𝑦)
151	12	rspcv 3573	. . . . . . . . . . . . 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 4713	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 ∈ 𝑉 → 𝑏 ∈ {𝑎, 𝑏})
154	153	ad2antll 729	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → 𝑏 ∈ {𝑎, 𝑏})
155	154	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑦 = {𝑎, 𝑏})) → 𝑏 ∈ {𝑎, 𝑏})
156		eleq2 2817	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = {𝑎, 𝑏} → (𝑏 ∈ 𝑦 ↔ 𝑏 ∈ {𝑎, 𝑏}))
157	156	ad2antll 729	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑦 = {𝑎, 𝑏})) → (𝑏 ∈ 𝑦 ↔ 𝑏 ∈ {𝑎, 𝑏}))
158	155, 157	mpbird 257	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑦 = {𝑎, 𝑏})) → 𝑏 ∈ 𝑦)
159	15	rspcv 3573	. . . . . . . . . . . . . 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 777	. . . . . . . . . . . . . . 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 2751	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐴) → 𝑎 = 𝑏)
163		eqneqall 2936	. . . . . . . . . . . . . . . . . . . . . . . . 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 728	. . . . . . . . . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . . . . . . . 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 413	. . . . . . . . . . . . . . . . . . . 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 857	. . . . . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = 𝐴 → (𝑏 = 𝐵 → ((((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑦 = {𝑎, 𝑏})) → {𝑎, 𝑏} = {𝐴, 𝐵})))
175		eqtr3 2751	. . . . . . . . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . . . . . . . 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 857	. . . . . . . . . . . . . . . . . . 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 857	. . . . . . . . . . . . . . . . 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 406	. . . . . . . . . . . . . . . 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 407	. . . . . . . . . . . . . . 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 2765	. . . . . . . . . . . . . 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 420	. . . . . . . . . . . . 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 412	. . . . . . . . . 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 3186	. . . . . . . . 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 242	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → (𝑦 ∈ 𝑃 → (∀𝑥 ∈ 𝑦 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → {𝐴, 𝐵} = 𝑦)))
190	189	imp 406	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ 𝑦 ∈ 𝑃) → (∀𝑥 ∈ 𝑦 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → {𝐴, 𝐵} = 𝑦))
191	190	ralrimiva 3121	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → ∀𝑦 ∈ 𝑃 (∀𝑥 ∈ 𝑦 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → {𝐴, 𝐵} = 𝑦))
192	141, 191	jca 511	. . . . 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 3579	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → ∃𝑝 ∈ 𝑃 (∀𝑥 ∈ 𝑝 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ∧ ∀𝑦 ∈ 𝑃 (∀𝑥 ∈ 𝑦 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → 𝑝 = 𝑦)))
194		raleq 3286	. . . . 5 ⊢ (𝑝 = 𝑦 → (∀𝑥 ∈ 𝑝 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ↔ ∀𝑥 ∈ 𝑦 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵)))
195	194	reu8 3693	. . . 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 234	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → ∃!𝑝 ∈ 𝑃 ∀𝑥 ∈ 𝑝 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵))
197	196	ex 412	. 2 ⊢ (𝜑 → (𝐴 ≠ 𝐵 → ∃!𝑝 ∈ 𝑃 ∀𝑥 ∈ 𝑝 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵)))
198	122, 197	impbid 212	1 ⊢ (𝜑 → (∃!𝑝 ∈ 𝑃 ∀𝑥 ∈ 𝑝 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ↔ 𝐴 ≠ 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044  ∃wrex 3053  ∃!wreu 3341  {crab 3394  Vcvv 3436  𝒫 cpw 4551  {cpr 4579  ‘cfv 6482  2c2 12183  ♯chash 14237

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-int 4897  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-1st 7924  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-1o 8388  df-2o 8389  df-oadd 8392  df-er 8625  df-en 8873  df-dom 8874  df-sdom 8875  df-fin 8876  df-dju 9797  df-card 9835  df-pnf 11151  df-mnf 11152  df-xr 11153  df-ltxr 11154  df-le 11155  df-sub 11349  df-neg 11350  df-nn 12129  df-2 12191  df-n0 12385  df-z 12472  df-uz 12736  df-fz 13411  df-hash 14238

This theorem is referenced by:  requad2  47617