Theorem paireqne 48183

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 3326	. . . 4 ⊢ (𝑝 = 𝑞 → (∀𝑥 ∈ 𝑝 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ↔ ∀𝑥 ∈ 𝑞 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵)))
2	1	reu8 3705	. . 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 2861	. . . . . . 7 ⊢ (𝑝 ∈ 𝑃 ↔ 𝑝 ∈ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2})
5		elss2prb 14525	. . . . . . 7 ⊢ (𝑝 ∈ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} ↔ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏}))
6	4, 5	bitri 278	. . . . . 6 ⊢ (𝑝 ∈ 𝑃 ↔ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏}))
7		raleq 3326	. . . . . . . . . . . 12 ⊢ (𝑝 = {𝑎, 𝑏} → (∀𝑥 ∈ 𝑝 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ↔ ∀𝑥 ∈ {𝑎, 𝑏} (𝑥 = 𝐴 ∨ 𝑥 = 𝐵)))
8		vex 3467	. . . . . . . . . . . . 13 ⊢ 𝑎 ∈ V
9		vex 3467	. . . . . . . . . . . . 13 ⊢ 𝑏 ∈ V
10		eqeq1 2773	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑎 → (𝑥 = 𝐴 ↔ 𝑎 = 𝐴))
11		eqeq1 2773	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑎 → (𝑥 = 𝐵 ↔ 𝑎 = 𝐵))
12	10, 11	orbi12d 931	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑎 → ((𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ↔ (𝑎 = 𝐴 ∨ 𝑎 = 𝐵)))
13		eqeq1 2773	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑏 → (𝑥 = 𝐴 ↔ 𝑏 = 𝐴))
14		eqeq1 2773	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑏 → (𝑥 = 𝐵 ↔ 𝑏 = 𝐵))
15	13, 14	orbi12d 931	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑏 → ((𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ↔ (𝑏 = 𝐴 ∨ 𝑏 = 𝐵)))
16	8, 9, 12, 15	ralpr 4671	. . . . . . . . . . . 12 ⊢ (∀𝑥 ∈ {𝑎, 𝑏} (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ↔ ((𝑎 = 𝐴 ∨ 𝑎 = 𝐵) ∧ (𝑏 = 𝐴 ∨ 𝑏 = 𝐵)))
17	7, 16	bitrdi 290	. . . . . . . . . . 11 ⊢ (𝑝 = {𝑎, 𝑏} → (∀𝑥 ∈ 𝑝 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ↔ ((𝑎 = 𝐴 ∨ 𝑎 = 𝐵) ∧ (𝑏 = 𝐴 ∨ 𝑏 = 𝐵))))
18		eqeq1 2773	. . . . . . . . . . . . 13 ⊢ (𝑝 = {𝑎, 𝑏} → (𝑝 = 𝑞 ↔ {𝑎, 𝑏} = 𝑞))
19	18	imbi2d 343	. . . . . . . . . . . 12 ⊢ (𝑝 = {𝑎, 𝑏} → ((∀𝑥 ∈ 𝑞 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → 𝑝 = 𝑞) ↔ (∀𝑥 ∈ 𝑞 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → {𝑎, 𝑏} = 𝑞)))
20	19	ralbidv 3194	. . . . . . . . . . 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 643	. . . . . . . . . 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 741	. . . . . . . . 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 520	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉))
26		prelpwi 5429	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → {𝐴, 𝐵} ∈ 𝒫 𝑉)
27	25, 26	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → {𝐴, 𝐵} ∈ 𝒫 𝑉)
28	27	ad3antrrr 742	. . . . . . . . . . . . . . 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 14431	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → (𝑎 ≠ 𝑏 ↔ (♯‘{𝑎, 𝑏}) = 2))
30	29	adantl 486	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → (𝑎 ≠ 𝑏 ↔ (♯‘{𝑎, 𝑏}) = 2))
31	30	biimpd 232	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → (𝑎 ≠ 𝑏 → (♯‘{𝑎, 𝑏}) = 2))
32	31	com12 33	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 ≠ 𝑏 → ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → (♯‘{𝑎, 𝑏}) = 2))
33	32	adantr 485	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏}) → ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → (♯‘{𝑎, 𝑏}) = 2))
34	33	impcom 412	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})) → (♯‘{𝑎, 𝑏}) = 2)
35	34	adantr 485	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})) ∧ ((𝑎 = 𝐴 ∨ 𝑎 = 𝐵) ∧ (𝑏 = 𝐴 ∨ 𝑏 = 𝐵))) → (♯‘{𝑎, 𝑏}) = 2)
36		eqtr3 2791	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑏 = 𝐴 ∧ 𝑎 = 𝐴) → 𝑏 = 𝑎)
37		eqneqall 2975	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑎 = 𝑏 → (𝑎 ≠ 𝑏 → (𝑝 = {𝑎, 𝑏} → {𝐴, 𝐵} = {𝑎, 𝑏})))
38	37	impd 415	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑎 = 𝑏 → ((𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏}) → {𝐴, 𝐵} = {𝑎, 𝑏}))
39	38	a1d 26	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑎 = 𝑏 → ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ((𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏}) → {𝐴, 𝐵} = {𝑎, 𝑏})))
40	39	impd 415	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑎 = 𝑏 → (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})) → {𝐴, 𝐵} = {𝑎, 𝑏}))
41	40	equcoms 2047	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑏 = 𝑎 → (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})) → {𝐴, 𝐵} = {𝑎, 𝑏}))
42	36, 41	syl 18	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑏 = 𝐴 ∧ 𝑎 = 𝐴) → (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})) → {𝐴, 𝐵} = {𝑎, 𝑏}))
43	42	ex 417	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑏 = 𝐴 → (𝑎 = 𝐴 → (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})) → {𝐴, 𝐵} = {𝑎, 𝑏})))
44		preq12 4706	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → {𝑎, 𝑏} = {𝐴, 𝐵})
45	44	eqcomd 2775	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → {𝐴, 𝐵} = {𝑎, 𝑏})
46	45	a1d 26	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})) → {𝐴, 𝐵} = {𝑎, 𝑏}))
47	46	expcom 418	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑏 = 𝐵 → (𝑎 = 𝐴 → (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})) → {𝐴, 𝐵} = {𝑎, 𝑏})))
48	43, 47	jaoi 870	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑏 = 𝐴 ∨ 𝑏 = 𝐵) → (𝑎 = 𝐴 → (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})) → {𝐴, 𝐵} = {𝑎, 𝑏})))
49	48	com12 33	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = 𝐴 → ((𝑏 = 𝐴 ∨ 𝑏 = 𝐵) → (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})) → {𝐴, 𝐵} = {𝑎, 𝑏})))
50		prcom 4703	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ {𝑎, 𝑏} = {𝑏, 𝑎}
51		preq12 4706	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑏 = 𝐴 ∧ 𝑎 = 𝐵) → {𝑏, 𝑎} = {𝐴, 𝐵})
52	50, 51	eqtrid 2816	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑏 = 𝐴 ∧ 𝑎 = 𝐵) → {𝑎, 𝑏} = {𝐴, 𝐵})
53	52	eqcomd 2775	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑏 = 𝐴 ∧ 𝑎 = 𝐵) → {𝐴, 𝐵} = {𝑎, 𝑏})
54	53	a1d 26	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑏 = 𝐴 ∧ 𝑎 = 𝐵) → (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})) → {𝐴, 𝐵} = {𝑎, 𝑏}))
55	54	ex 417	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑏 = 𝐴 → (𝑎 = 𝐵 → (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})) → {𝐴, 𝐵} = {𝑎, 𝑏})))
56		eqtr3 2791	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑏 = 𝐵 ∧ 𝑎 = 𝐵) → 𝑏 = 𝑎)
57	56, 41	syl 18	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑏 = 𝐵 ∧ 𝑎 = 𝐵) → (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})) → {𝐴, 𝐵} = {𝑎, 𝑏}))
58	57	ex 417	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑏 = 𝐵 → (𝑎 = 𝐵 → (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})) → {𝐴, 𝐵} = {𝑎, 𝑏})))
59	55, 58	jaoi 870	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑏 = 𝐴 ∨ 𝑏 = 𝐵) → (𝑎 = 𝐵 → (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})) → {𝐴, 𝐵} = {𝑎, 𝑏})))
60	59	com12 33	. . . . . . . . . . . . . . . . . . . 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 870	. . . . . . . . . . . . . . . . . . 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 411	. . . . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . . . . 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 6890	. . . . . . . . . . . . . . . 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 260	. . . . . . . . . . . . . . 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 520	. . . . . . . . . . . . . 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 2861	. . . . . . . . . . . . . . 15 ⊢ ({𝐴, 𝐵} ∈ 𝑃 ↔ {𝐴, 𝐵} ∈ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2})
68		fveqeq2 6891	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = {𝐴, 𝐵} → ((♯‘𝑥) = 2 ↔ (♯‘{𝐴, 𝐵}) = 2))
69	68	elrab 3659	. . . . . . . . . . . . . . 15 ⊢ ({𝐴, 𝐵} ∈ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} ↔ ({𝐴, 𝐵} ∈ 𝒫 𝑉 ∧ (♯‘{𝐴, 𝐵}) = 2))
70	67, 69	bitri 278	. . . . . . . . . . . . . 14 ⊢ ({𝐴, 𝐵} ∈ 𝑃 ↔ ({𝐴, 𝐵} ∈ 𝒫 𝑉 ∧ (♯‘{𝐴, 𝐵}) = 2))
71	66, 70	sylibr 237	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})) ∧ ((𝑎 = 𝐴 ∨ 𝑎 = 𝐵) ∧ (𝑏 = 𝐴 ∨ 𝑏 = 𝐵))) → {𝐴, 𝐵} ∈ 𝑃)
72		raleq 3326	. . . . . . . . . . . . . . 15 ⊢ (𝑞 = {𝐴, 𝐵} → (∀𝑥 ∈ 𝑞 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ↔ ∀𝑥 ∈ {𝐴, 𝐵} (𝑥 = 𝐴 ∨ 𝑥 = 𝐵)))
73		eqeq2 2781	. . . . . . . . . . . . . . 15 ⊢ (𝑞 = {𝐴, 𝐵} → ({𝑎, 𝑏} = 𝑞 ↔ {𝑎, 𝑏} = {𝐴, 𝐵}))
74	72, 73	imbi12d 347	. . . . . . . . . . . . . 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 3586	. . . . . . . . . . . . 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 18	. . . . . . . . . . . 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 2773	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝐴 ↔ 𝐴 = 𝐴))
78		eqeq1 2773	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝐵 ↔ 𝐴 = 𝐵))
79	77, 78	orbi12d 931	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝐴 → ((𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ↔ (𝐴 = 𝐴 ∨ 𝐴 = 𝐵)))
80		eqeq1 2773	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝐵 → (𝑥 = 𝐴 ↔ 𝐵 = 𝐴))
81		eqeq1 2773	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝐵 → (𝑥 = 𝐵 ↔ 𝐵 = 𝐵))
82	80, 81	orbi12d 931	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝐵 → ((𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ↔ (𝐵 = 𝐴 ∨ 𝐵 = 𝐵)))
83	79, 82	ralprg 4667	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (∀𝑥 ∈ {𝐴, 𝐵} (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ↔ ((𝐴 = 𝐴 ∨ 𝐴 = 𝐵) ∧ (𝐵 = 𝐴 ∨ 𝐵 = 𝐵))))
84	25, 83	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (∀𝑥 ∈ {𝐴, 𝐵} (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ↔ ((𝐴 = 𝐴 ∨ 𝐴 = 𝐵) ∧ (𝐵 = 𝐴 ∨ 𝐵 = 𝐵))))
85	84	imbi1d 344	. . . . . . . . . . . . . 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 742	. . . . . . . . . . . . 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 2769	. . . . . . . . . . . . . . . 16 ⊢ 𝐴 = 𝐴
88	87	orci 878	. . . . . . . . . . . . . . 15 ⊢ (𝐴 = 𝐴 ∨ 𝐴 = 𝐵)
89		eqid 2769	. . . . . . . . . . . . . . . 16 ⊢ 𝐵 = 𝐵
90	89	olci 879	. . . . . . . . . . . . . . 15 ⊢ (𝐵 = 𝐴 ∨ 𝐵 = 𝐵)
91		pm5.5 364	. . . . . . . . . . . . . . 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 704	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 = 𝐴 ∨ 𝐴 = 𝐵) ∧ (𝐵 = 𝐴 ∨ 𝐵 = 𝐵)) → {𝑎, 𝑏} = {𝐴, 𝐵}) ↔ {𝑎, 𝑏} = {𝐴, 𝐵})
93	8, 9	pm3.2i 475	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 ∈ V ∧ 𝑏 ∈ V)
94		preq12bg 4822	. . . . . . . . . . . . . . . . . . 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 598	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ({𝑎, 𝑏} = {𝐴, 𝐵} ↔ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) ∨ (𝑎 = 𝐵 ∧ 𝑏 = 𝐴))))
96	95	adantr 485	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ({𝑎, 𝑏} = {𝐴, 𝐵} ↔ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) ∨ (𝑎 = 𝐵 ∧ 𝑏 = 𝐴))))
97	96	adantr 485	. . . . . . . . . . . . . . . 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 2786	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (𝑎 = 𝑏 ↔ 𝐴 = 𝐵))
99	98	necon3bid 3008	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (𝑎 ≠ 𝑏 ↔ 𝐴 ≠ 𝐵))
100	99	biimpd 232	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (𝑎 ≠ 𝑏 → 𝐴 ≠ 𝐵))
101		eqeq12 2786	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑎 = 𝐵 ∧ 𝑏 = 𝐴) → (𝑎 = 𝑏 ↔ 𝐵 = 𝐴))
102	101	necon3bid 3008	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑎 = 𝐵 ∧ 𝑏 = 𝐴) → (𝑎 ≠ 𝑏 ↔ 𝐵 ≠ 𝐴))
103	102	biimpd 232	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑎 = 𝐵 ∧ 𝑏 = 𝐴) → (𝑎 ≠ 𝑏 → 𝐵 ≠ 𝐴))
104		necom 3017	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐴 ≠ 𝐵 ↔ 𝐵 ≠ 𝐴)
105	103, 104	imbitrrdi 255	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑎 = 𝐵 ∧ 𝑏 = 𝐴) → (𝑎 ≠ 𝑏 → 𝐴 ≠ 𝐵))
106	100, 105	jaoi 870	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) ∨ (𝑎 = 𝐵 ∧ 𝑏 = 𝐴)) → (𝑎 ≠ 𝑏 → 𝐴 ≠ 𝐵))
107	106	com12 33	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 ≠ 𝑏 → (((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) ∨ (𝑎 = 𝐵 ∧ 𝑏 = 𝐴)) → 𝐴 ≠ 𝐵))
108	107	ad2antrl 740	. . . . . . . . . . . . . . . 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 243	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})) → ({𝑎, 𝑏} = {𝐴, 𝐵} → 𝐴 ≠ 𝐵))
110	109	adantr 485	. . . . . . . . . . . . . 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 245	. . . . . . . . . . . . 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 243	. . . . . . . . . . . 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 48	. . . . . . . . . . 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 417	. . . . . . . . . 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 415	. . . . . . . . 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 243	. . . . . . . 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 417	. . . . . . 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 3228	. . . . . 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 245	. . . . 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 411	. . . 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 3172	. . 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 245	. 2 ⊢ (𝜑 → (∃!𝑝 ∈ 𝑃 ∀𝑥 ∈ 𝑝 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → 𝐴 ≠ 𝐵))
123	27	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → {𝐴, 𝐵} ∈ 𝒫 𝑉)
124		hashprg 14431	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴 ≠ 𝐵 ↔ (♯‘{𝐴, 𝐵}) = 2))
125	25, 124	syl 18	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 ≠ 𝐵 ↔ (♯‘{𝐴, 𝐵}) = 2))
126	125	biimpd 232	. . . . . . . 8 ⊢ (𝜑 → (𝐴 ≠ 𝐵 → (♯‘{𝐴, 𝐵}) = 2))
127	126	imp 411	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → (♯‘{𝐴, 𝐵}) = 2)
128	123, 127	jca 520	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → ({𝐴, 𝐵} ∈ 𝒫 𝑉 ∧ (♯‘{𝐴, 𝐵}) = 2))
129	128, 70	sylibr 237	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → {𝐴, 𝐵} ∈ 𝑃)
130		raleq 3326	. . . . . . 7 ⊢ (𝑝 = {𝐴, 𝐵} → (∀𝑥 ∈ 𝑝 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ↔ ∀𝑥 ∈ {𝐴, 𝐵} (𝑥 = 𝐴 ∨ 𝑥 = 𝐵)))
131		eqeq1 2773	. . . . . . . . 9 ⊢ (𝑝 = {𝐴, 𝐵} → (𝑝 = 𝑦 ↔ {𝐴, 𝐵} = 𝑦))
132	131	imbi2d 343	. . . . . . . 8 ⊢ (𝑝 = {𝐴, 𝐵} → ((∀𝑥 ∈ 𝑦 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → 𝑝 = 𝑦) ↔ (∀𝑥 ∈ 𝑦 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → {𝐴, 𝐵} = 𝑦)))
133	132	ralbidv 3194	. . . . . . 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 643	. . . . . 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 486	. . . . 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 3467	. . . . . . . . 9 ⊢ 𝑥 ∈ V
137	136	elpr 4619	. . . . . . . 8 ⊢ (𝑥 ∈ {𝐴, 𝐵} ↔ (𝑥 = 𝐴 ∨ 𝑥 = 𝐵))
138	137	bilani 509	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ 𝑥 ∈ {𝐴, 𝐵}) → (𝑥 = 𝐴 ∨ 𝑥 = 𝐵))
139	138	ralrimiva 3163	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → ∀𝑥 ∈ {𝐴, 𝐵} (𝑥 = 𝐴 ∨ 𝑥 = 𝐵))
140	3	eleq2i 2861	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝑃 ↔ 𝑦 ∈ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2})
141		elss2prb 14525	. . . . . . . . . 10 ⊢ (𝑦 ∈ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} ↔ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝑦 = {𝑎, 𝑏}))
142	140, 141	bitri 278	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝑃 ↔ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝑦 = {𝑎, 𝑏}))
143		prid1g 4731	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 ∈ 𝑉 → 𝑎 ∈ {𝑎, 𝑏})
144	143	ad2antrl 740	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → 𝑎 ∈ {𝑎, 𝑏})
145	144	adantr 485	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑦 = {𝑎, 𝑏})) → 𝑎 ∈ {𝑎, 𝑏})
146		eleq2 2858	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = {𝑎, 𝑏} → (𝑎 ∈ 𝑦 ↔ 𝑎 ∈ {𝑎, 𝑏}))
147	146	ad2antll 741	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑦 = {𝑎, 𝑏})) → (𝑎 ∈ 𝑦 ↔ 𝑎 ∈ {𝑎, 𝑏}))
148	145, 147	mpbird 260	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑦 = {𝑎, 𝑏})) → 𝑎 ∈ 𝑦)
149	12	rspcv 3586	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ 𝑦 → (∀𝑥 ∈ 𝑦 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → (𝑎 = 𝐴 ∨ 𝑎 = 𝐵)))
150	148, 149	syl 18	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑦 = {𝑎, 𝑏})) → (∀𝑥 ∈ 𝑦 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → (𝑎 = 𝐴 ∨ 𝑎 = 𝐵)))
151		prid2g 4732	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 ∈ 𝑉 → 𝑏 ∈ {𝑎, 𝑏})
152	151	ad2antll 741	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → 𝑏 ∈ {𝑎, 𝑏})
153	152	adantr 485	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑦 = {𝑎, 𝑏})) → 𝑏 ∈ {𝑎, 𝑏})
154		eleq2 2858	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = {𝑎, 𝑏} → (𝑏 ∈ 𝑦 ↔ 𝑏 ∈ {𝑎, 𝑏}))
155	154	ad2antll 741	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑦 = {𝑎, 𝑏})) → (𝑏 ∈ 𝑦 ↔ 𝑏 ∈ {𝑎, 𝑏}))
156	153, 155	mpbird 260	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑦 = {𝑎, 𝑏})) → 𝑏 ∈ 𝑦)
157	15	rspcv 3586	. . . . . . . . . . . . . 14 ⊢ (𝑏 ∈ 𝑦 → (∀𝑥 ∈ 𝑦 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → (𝑏 = 𝐴 ∨ 𝑏 = 𝐵)))
158	156, 157	syl 18	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑦 = {𝑎, 𝑏})) → (∀𝑥 ∈ 𝑦 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → (𝑏 = 𝐴 ∨ 𝑏 = 𝐵)))
159		simplrr 789	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑦 = {𝑎, 𝑏})) ∧ ((𝑏 = 𝐴 ∨ 𝑏 = 𝐵) ∧ (𝑎 = 𝐴 ∨ 𝑎 = 𝐵))) → 𝑦 = {𝑎, 𝑏})
160		eqtr3 2791	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐴) → 𝑎 = 𝑏)
161		eqneqall 2975	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑎 = 𝑏 → (𝑎 ≠ 𝑏 → {𝑎, 𝑏} = {𝐴, 𝐵}))
162	161	com12 33	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑎 ≠ 𝑏 → (𝑎 = 𝑏 → {𝑎, 𝑏} = {𝐴, 𝐵}))
163	162	ad2antrl 740	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑦 = {𝑎, 𝑏})) → (𝑎 = 𝑏 → {𝑎, 𝑏} = {𝐴, 𝐵}))
164	163	com12 33	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑎 = 𝑏 → ((((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑦 = {𝑎, 𝑏})) → {𝑎, 𝑏} = {𝐴, 𝐵}))
165	160, 164	syl 18	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐴) → ((((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑦 = {𝑎, 𝑏})) → {𝑎, 𝑏} = {𝐴, 𝐵}))
166	165	ex 417	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = 𝐴 → (𝑏 = 𝐴 → ((((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑦 = {𝑎, 𝑏})) → {𝑎, 𝑏} = {𝐴, 𝐵})))
167	52	a1d 26	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑏 = 𝐴 ∧ 𝑎 = 𝐵) → ((((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑦 = {𝑎, 𝑏})) → {𝑎, 𝑏} = {𝐴, 𝐵}))
168	167	expcom 418	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = 𝐵 → (𝑏 = 𝐴 → ((((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑦 = {𝑎, 𝑏})) → {𝑎, 𝑏} = {𝐴, 𝐵})))
169	166, 168	jaoi 870	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑎 = 𝐴 ∨ 𝑎 = 𝐵) → (𝑏 = 𝐴 → ((((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑦 = {𝑎, 𝑏})) → {𝑎, 𝑏} = {𝐴, 𝐵})))
170	169	com12 33	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = 𝐴 → ((𝑎 = 𝐴 ∨ 𝑎 = 𝐵) → ((((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑦 = {𝑎, 𝑏})) → {𝑎, 𝑏} = {𝐴, 𝐵})))
171	44	a1d 26	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → ((((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑦 = {𝑎, 𝑏})) → {𝑎, 𝑏} = {𝐴, 𝐵}))
172	171	ex 417	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = 𝐴 → (𝑏 = 𝐵 → ((((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑦 = {𝑎, 𝑏})) → {𝑎, 𝑏} = {𝐴, 𝐵})))
173		eqtr3 2791	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑎 = 𝐵 ∧ 𝑏 = 𝐵) → 𝑎 = 𝑏)
174	173, 164	syl 18	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑎 = 𝐵 ∧ 𝑏 = 𝐵) → ((((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑦 = {𝑎, 𝑏})) → {𝑎, 𝑏} = {𝐴, 𝐵}))
175	174	ex 417	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = 𝐵 → (𝑏 = 𝐵 → ((((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑦 = {𝑎, 𝑏})) → {𝑎, 𝑏} = {𝐴, 𝐵})))
176	172, 175	jaoi 870	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑎 = 𝐴 ∨ 𝑎 = 𝐵) → (𝑏 = 𝐵 → ((((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑦 = {𝑎, 𝑏})) → {𝑎, 𝑏} = {𝐴, 𝐵})))
177	176	com12 33	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = 𝐵 → ((𝑎 = 𝐴 ∨ 𝑎 = 𝐵) → ((((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑦 = {𝑎, 𝑏})) → {𝑎, 𝑏} = {𝐴, 𝐵})))
178	170, 177	jaoi 870	. . . . . . . . . . . . . . . . 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})))
179	178	imp 411	. . . . . . . . . . . . . . . 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}))
180	179	impcom 412	. . . . . . . . . . . . . . 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})
181	159, 180	eqtr2d 2805	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑦 = {𝑎, 𝑏})) ∧ ((𝑏 = 𝐴 ∨ 𝑏 = 𝐵) ∧ (𝑎 = 𝐴 ∨ 𝑎 = 𝐵))) → {𝐴, 𝐵} = 𝑦)
182	181	exp32 425	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑦 = {𝑎, 𝑏})) → ((𝑏 = 𝐴 ∨ 𝑏 = 𝐵) → ((𝑎 = 𝐴 ∨ 𝑎 = 𝐵) → {𝐴, 𝐵} = 𝑦)))
183	158, 182	syld 48	. . . . . . . . . . . 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)))
184	150, 183	mpdd 44	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑦 = {𝑎, 𝑏})) → (∀𝑥 ∈ 𝑦 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → {𝐴, 𝐵} = 𝑦))
185	184	ex 417	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ((𝑎 ≠ 𝑏 ∧ 𝑦 = {𝑎, 𝑏}) → (∀𝑥 ∈ 𝑦 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → {𝐴, 𝐵} = 𝑦)))
186	185	rexlimdvva 3228	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → (∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝑦 = {𝑎, 𝑏}) → (∀𝑥 ∈ 𝑦 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → {𝐴, 𝐵} = 𝑦)))
187	142, 186	biimtrid 245	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → (𝑦 ∈ 𝑃 → (∀𝑥 ∈ 𝑦 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → {𝐴, 𝐵} = 𝑦)))
188	187	imp 411	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ 𝑦 ∈ 𝑃) → (∀𝑥 ∈ 𝑦 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → {𝐴, 𝐵} = 𝑦))
189	188	ralrimiva 3163	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → ∀𝑦 ∈ 𝑃 (∀𝑥 ∈ 𝑦 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → {𝐴, 𝐵} = 𝑦))
190	139, 189	jca 520	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → (∀𝑥 ∈ {𝐴, 𝐵} (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ∧ ∀𝑦 ∈ 𝑃 (∀𝑥 ∈ 𝑦 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → {𝐴, 𝐵} = 𝑦)))
191	129, 135, 190	rspcedvd 3592	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → ∃𝑝 ∈ 𝑃 (∀𝑥 ∈ 𝑝 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ∧ ∀𝑦 ∈ 𝑃 (∀𝑥 ∈ 𝑦 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → 𝑝 = 𝑦)))
192		raleq 3326	. . . . 5 ⊢ (𝑝 = 𝑦 → (∀𝑥 ∈ 𝑝 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ↔ ∀𝑥 ∈ 𝑦 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵)))
193	192	reu8 3705	. . . 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)))
194	191, 193	sylibr 237	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → ∃!𝑝 ∈ 𝑃 ∀𝑥 ∈ 𝑝 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵))
195	194	ex 417	. 2 ⊢ (𝜑 → (𝐴 ≠ 𝐵 → ∃!𝑝 ∈ 𝑃 ∀𝑥 ∈ 𝑝 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵)))
196	122, 195	impbid 215	1 ⊢ (𝜑 → (∃!𝑝 ∈ 𝑃 ∀𝑥 ∈ 𝑝 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ↔ 𝐴 ≠ 𝐵))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   = wceq 1567   ∈ wcel 2149   ≠ wne 2964  ∀wral 3085  ∃wrex 3095  ∃!wreu 3374  {crab 3423  Vcvv 3463  𝒫 cpw 4567  {cpr 4596  ‘cfv 6537  2c2 12295  ♯chash 14366

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-cnex 11156  ax-resscn 11157  ax-1cn 11158  ax-icn 11159  ax-addcl 11160  ax-addrcl 11161  ax-mulcl 11162  ax-mulrcl 11163  ax-mulcom 11164  ax-addass 11165  ax-mulass 11166  ax-distr 11167  ax-i2m1 11168  ax-1ne0 11169  ax-1rid 11170  ax-rnegex 11171  ax-rrecex 11172  ax-cnre 11173  ax-pre-lttri 11174  ax-pre-lttrn 11175  ax-pre-ltadd 11176  ax-pre-mulgt0 11177

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-int 4917  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7863  df-1st 7986  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-1o 8453  df-2o 8454  df-oadd 8457  df-er 8694  df-en 8944  df-dom 8945  df-sdom 8946  df-fin 8947  df-dju 9887  df-card 9925  df-pnf 11245  df-mnf 11246  df-xr 11247  df-ltxr 11248  df-le 11249  df-sub 11443  df-neg 11444  df-nn 12234  df-2 12303  df-n0 12505  df-z 12592  df-uz 12863  df-fz 13536  df-hash 14367

This theorem is referenced by:  requad2  48311