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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 (𝜑𝐴𝑉)
paireqne.b (𝜑𝐵𝑉)
paireqne.p 𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}
Assertion
Ref Expression
paireqne (𝜑 → (∃!𝑝𝑃𝑥𝑝 (𝑥 = 𝐴𝑥 = 𝐵) ↔ 𝐴𝐵))
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.
StepHypRef Expression
1 raleq 3329 . . . 4 (𝑝 = 𝑞 → (∀𝑥𝑝 (𝑥 = 𝐴𝑥 = 𝐵) ↔ ∀𝑥𝑞 (𝑥 = 𝐴𝑥 = 𝐵)))
21reu8 3613 . . 3 (∃!𝑝𝑃𝑥𝑝 (𝑥 = 𝐴𝑥 = 𝐵) ↔ ∃𝑝𝑃 (∀𝑥𝑝 (𝑥 = 𝐴𝑥 = 𝐵) ∧ ∀𝑞𝑃 (∀𝑥𝑞 (𝑥 = 𝐴𝑥 = 𝐵) → 𝑝 = 𝑞)))
3 paireqne.p . . . . . . . 8 𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}
43eleq2i 2850 . . . . . . 7 (𝑝𝑃𝑝 ∈ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2})
5 elss2prb 13583 . . . . . . 7 (𝑝 ∈ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} ↔ ∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝑝 = {𝑎, 𝑏}))
64, 5bitri 267 . . . . . 6 (𝑝𝑃 ↔ ∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝑝 = {𝑎, 𝑏}))
7 raleq 3329 . . . . . . . . . . . 12 (𝑝 = {𝑎, 𝑏} → (∀𝑥𝑝 (𝑥 = 𝐴𝑥 = 𝐵) ↔ ∀𝑥 ∈ {𝑎, 𝑏} (𝑥 = 𝐴𝑥 = 𝐵)))
8 vex 3400 . . . . . . . . . . . . 13 𝑎 ∈ V
9 vex 3400 . . . . . . . . . . . . 13 𝑏 ∈ V
10 eqeq1 2781 . . . . . . . . . . . . . 14 (𝑥 = 𝑎 → (𝑥 = 𝐴𝑎 = 𝐴))
11 eqeq1 2781 . . . . . . . . . . . . . 14 (𝑥 = 𝑎 → (𝑥 = 𝐵𝑎 = 𝐵))
1210, 11orbi12d 905 . . . . . . . . . . . . 13 (𝑥 = 𝑎 → ((𝑥 = 𝐴𝑥 = 𝐵) ↔ (𝑎 = 𝐴𝑎 = 𝐵)))
13 eqeq1 2781 . . . . . . . . . . . . . 14 (𝑥 = 𝑏 → (𝑥 = 𝐴𝑏 = 𝐴))
14 eqeq1 2781 . . . . . . . . . . . . . 14 (𝑥 = 𝑏 → (𝑥 = 𝐵𝑏 = 𝐵))
1513, 14orbi12d 905 . . . . . . . . . . . . 13 (𝑥 = 𝑏 → ((𝑥 = 𝐴𝑥 = 𝐵) ↔ (𝑏 = 𝐴𝑏 = 𝐵)))
168, 9, 12, 15ralpr 4466 . . . . . . . . . . . 12 (∀𝑥 ∈ {𝑎, 𝑏} (𝑥 = 𝐴𝑥 = 𝐵) ↔ ((𝑎 = 𝐴𝑎 = 𝐵) ∧ (𝑏 = 𝐴𝑏 = 𝐵)))
177, 16syl6bb 279 . . . . . . . . . . 11 (𝑝 = {𝑎, 𝑏} → (∀𝑥𝑝 (𝑥 = 𝐴𝑥 = 𝐵) ↔ ((𝑎 = 𝐴𝑎 = 𝐵) ∧ (𝑏 = 𝐴𝑏 = 𝐵))))
18 eqeq1 2781 . . . . . . . . . . . . 13 (𝑝 = {𝑎, 𝑏} → (𝑝 = 𝑞 ↔ {𝑎, 𝑏} = 𝑞))
1918imbi2d 332 . . . . . . . . . . . 12 (𝑝 = {𝑎, 𝑏} → ((∀𝑥𝑞 (𝑥 = 𝐴𝑥 = 𝐵) → 𝑝 = 𝑞) ↔ (∀𝑥𝑞 (𝑥 = 𝐴𝑥 = 𝐵) → {𝑎, 𝑏} = 𝑞)))
2019ralbidv 3167 . . . . . . . . . . 11 (𝑝 = {𝑎, 𝑏} → (∀𝑞𝑃 (∀𝑥𝑞 (𝑥 = 𝐴𝑥 = 𝐵) → 𝑝 = 𝑞) ↔ ∀𝑞𝑃 (∀𝑥𝑞 (𝑥 = 𝐴𝑥 = 𝐵) → {𝑎, 𝑏} = 𝑞)))
2117, 20anbi12d 624 . . . . . . . . . 10 (𝑝 = {𝑎, 𝑏} → ((∀𝑥𝑝 (𝑥 = 𝐴𝑥 = 𝐵) ∧ ∀𝑞𝑃 (∀𝑥𝑞 (𝑥 = 𝐴𝑥 = 𝐵) → 𝑝 = 𝑞)) ↔ (((𝑎 = 𝐴𝑎 = 𝐵) ∧ (𝑏 = 𝐴𝑏 = 𝐵)) ∧ ∀𝑞𝑃 (∀𝑥𝑞 (𝑥 = 𝐴𝑥 = 𝐵) → {𝑎, 𝑏} = 𝑞))))
2221ad2antll 719 . . . . . . . . 9 (((𝜑 ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑝 = {𝑎, 𝑏})) → ((∀𝑥𝑝 (𝑥 = 𝐴𝑥 = 𝐵) ∧ ∀𝑞𝑃 (∀𝑥𝑞 (𝑥 = 𝐴𝑥 = 𝐵) → 𝑝 = 𝑞)) ↔ (((𝑎 = 𝐴𝑎 = 𝐵) ∧ (𝑏 = 𝐴𝑏 = 𝐵)) ∧ ∀𝑞𝑃 (∀𝑥𝑞 (𝑥 = 𝐴𝑥 = 𝐵) → {𝑎, 𝑏} = 𝑞))))
23 paireqne.a . . . . . . . . . . . . . . . . . 18 (𝜑𝐴𝑉)
24 paireqne.b . . . . . . . . . . . . . . . . . 18 (𝜑𝐵𝑉)
2523, 24jca 507 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝐴𝑉𝐵𝑉))
26 prelpwi 5147 . . . . . . . . . . . . . . . . 17 ((𝐴𝑉𝐵𝑉) → {𝐴, 𝐵} ∈ 𝒫 𝑉)
2725, 26syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → {𝐴, 𝐵} ∈ 𝒫 𝑉)
2827ad3antrrr 720 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑝 = {𝑎, 𝑏})) ∧ ((𝑎 = 𝐴𝑎 = 𝐵) ∧ (𝑏 = 𝐴𝑏 = 𝐵))) → {𝐴, 𝐵} ∈ 𝒫 𝑉)
29 hashprg 13497 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑎𝑉𝑏𝑉) → (𝑎𝑏 ↔ (♯‘{𝑎, 𝑏}) = 2))
3029adantl 475 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑎𝑉𝑏𝑉)) → (𝑎𝑏 ↔ (♯‘{𝑎, 𝑏}) = 2))
3130biimpd 221 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑎𝑉𝑏𝑉)) → (𝑎𝑏 → (♯‘{𝑎, 𝑏}) = 2))
3231com12 32 . . . . . . . . . . . . . . . . . . 19 (𝑎𝑏 → ((𝜑 ∧ (𝑎𝑉𝑏𝑉)) → (♯‘{𝑎, 𝑏}) = 2))
3332adantr 474 . . . . . . . . . . . . . . . . . 18 ((𝑎𝑏𝑝 = {𝑎, 𝑏}) → ((𝜑 ∧ (𝑎𝑉𝑏𝑉)) → (♯‘{𝑎, 𝑏}) = 2))
3433impcom 398 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑝 = {𝑎, 𝑏})) → (♯‘{𝑎, 𝑏}) = 2)
3534adantr 474 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑝 = {𝑎, 𝑏})) ∧ ((𝑎 = 𝐴𝑎 = 𝐵) ∧ (𝑏 = 𝐴𝑏 = 𝐵))) → (♯‘{𝑎, 𝑏}) = 2)
36 eqtr3 2800 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑏 = 𝐴𝑎 = 𝐴) → 𝑏 = 𝑎)
37 eqneqall 2979 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑎 = 𝑏 → (𝑎𝑏 → (𝑝 = {𝑎, 𝑏} → {𝐴, 𝐵} = {𝑎, 𝑏})))
3837impd 400 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑎 = 𝑏 → ((𝑎𝑏𝑝 = {𝑎, 𝑏}) → {𝐴, 𝐵} = {𝑎, 𝑏}))
3938a1d 25 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑎 = 𝑏 → ((𝜑 ∧ (𝑎𝑉𝑏𝑉)) → ((𝑎𝑏𝑝 = {𝑎, 𝑏}) → {𝐴, 𝐵} = {𝑎, 𝑏})))
4039impd 400 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑎 = 𝑏 → (((𝜑 ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑝 = {𝑎, 𝑏})) → {𝐴, 𝐵} = {𝑎, 𝑏}))
4140equcoms 2066 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑏 = 𝑎 → (((𝜑 ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑝 = {𝑎, 𝑏})) → {𝐴, 𝐵} = {𝑎, 𝑏}))
4236, 41syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑏 = 𝐴𝑎 = 𝐴) → (((𝜑 ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑝 = {𝑎, 𝑏})) → {𝐴, 𝐵} = {𝑎, 𝑏}))
4342ex 403 . . . . . . . . . . . . . . . . . . . . . 22 (𝑏 = 𝐴 → (𝑎 = 𝐴 → (((𝜑 ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑝 = {𝑎, 𝑏})) → {𝐴, 𝐵} = {𝑎, 𝑏})))
44 preq12 4501 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑎 = 𝐴𝑏 = 𝐵) → {𝑎, 𝑏} = {𝐴, 𝐵})
4544eqcomd 2783 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑎 = 𝐴𝑏 = 𝐵) → {𝐴, 𝐵} = {𝑎, 𝑏})
4645a1d 25 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑎 = 𝐴𝑏 = 𝐵) → (((𝜑 ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑝 = {𝑎, 𝑏})) → {𝐴, 𝐵} = {𝑎, 𝑏}))
4746expcom 404 . . . . . . . . . . . . . . . . . . . . . 22 (𝑏 = 𝐵 → (𝑎 = 𝐴 → (((𝜑 ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑝 = {𝑎, 𝑏})) → {𝐴, 𝐵} = {𝑎, 𝑏})))
4843, 47jaoi 846 . . . . . . . . . . . . . . . . . . . . 21 ((𝑏 = 𝐴𝑏 = 𝐵) → (𝑎 = 𝐴 → (((𝜑 ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑝 = {𝑎, 𝑏})) → {𝐴, 𝐵} = {𝑎, 𝑏})))
4948com12 32 . . . . . . . . . . . . . . . . . . . 20 (𝑎 = 𝐴 → ((𝑏 = 𝐴𝑏 = 𝐵) → (((𝜑 ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑝 = {𝑎, 𝑏})) → {𝐴, 𝐵} = {𝑎, 𝑏})))
50 prcom 4498 . . . . . . . . . . . . . . . . . . . . . . . . . 26 {𝑎, 𝑏} = {𝑏, 𝑎}
51 preq12 4501 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑏 = 𝐴𝑎 = 𝐵) → {𝑏, 𝑎} = {𝐴, 𝐵})
5250, 51syl5eq 2825 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑏 = 𝐴𝑎 = 𝐵) → {𝑎, 𝑏} = {𝐴, 𝐵})
5352eqcomd 2783 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑏 = 𝐴𝑎 = 𝐵) → {𝐴, 𝐵} = {𝑎, 𝑏})
5453a1d 25 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑏 = 𝐴𝑎 = 𝐵) → (((𝜑 ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑝 = {𝑎, 𝑏})) → {𝐴, 𝐵} = {𝑎, 𝑏}))
5554ex 403 . . . . . . . . . . . . . . . . . . . . . 22 (𝑏 = 𝐴 → (𝑎 = 𝐵 → (((𝜑 ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑝 = {𝑎, 𝑏})) → {𝐴, 𝐵} = {𝑎, 𝑏})))
56 eqtr3 2800 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑏 = 𝐵𝑎 = 𝐵) → 𝑏 = 𝑎)
5756, 41syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑏 = 𝐵𝑎 = 𝐵) → (((𝜑 ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑝 = {𝑎, 𝑏})) → {𝐴, 𝐵} = {𝑎, 𝑏}))
5857ex 403 . . . . . . . . . . . . . . . . . . . . . 22 (𝑏 = 𝐵 → (𝑎 = 𝐵 → (((𝜑 ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑝 = {𝑎, 𝑏})) → {𝐴, 𝐵} = {𝑎, 𝑏})))
5955, 58jaoi 846 . . . . . . . . . . . . . . . . . . . . 21 ((𝑏 = 𝐴𝑏 = 𝐵) → (𝑎 = 𝐵 → (((𝜑 ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑝 = {𝑎, 𝑏})) → {𝐴, 𝐵} = {𝑎, 𝑏})))
6059com12 32 . . . . . . . . . . . . . . . . . . . 20 (𝑎 = 𝐵 → ((𝑏 = 𝐴𝑏 = 𝐵) → (((𝜑 ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑝 = {𝑎, 𝑏})) → {𝐴, 𝐵} = {𝑎, 𝑏})))
6149, 60jaoi 846 . . . . . . . . . . . . . . . . . . 19 ((𝑎 = 𝐴𝑎 = 𝐵) → ((𝑏 = 𝐴𝑏 = 𝐵) → (((𝜑 ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑝 = {𝑎, 𝑏})) → {𝐴, 𝐵} = {𝑎, 𝑏})))
6261imp 397 . . . . . . . . . . . . . . . . . 18 (((𝑎 = 𝐴𝑎 = 𝐵) ∧ (𝑏 = 𝐴𝑏 = 𝐵)) → (((𝜑 ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑝 = {𝑎, 𝑏})) → {𝐴, 𝐵} = {𝑎, 𝑏}))
6362impcom 398 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑝 = {𝑎, 𝑏})) ∧ ((𝑎 = 𝐴𝑎 = 𝐵) ∧ (𝑏 = 𝐴𝑏 = 𝐵))) → {𝐴, 𝐵} = {𝑎, 𝑏})
6463fveqeq2d 6454 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑝 = {𝑎, 𝑏})) ∧ ((𝑎 = 𝐴𝑎 = 𝐵) ∧ (𝑏 = 𝐴𝑏 = 𝐵))) → ((♯‘{𝐴, 𝐵}) = 2 ↔ (♯‘{𝑎, 𝑏}) = 2))
6535, 64mpbird 249 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑝 = {𝑎, 𝑏})) ∧ ((𝑎 = 𝐴𝑎 = 𝐵) ∧ (𝑏 = 𝐴𝑏 = 𝐵))) → (♯‘{𝐴, 𝐵}) = 2)
6628, 65jca 507 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑝 = {𝑎, 𝑏})) ∧ ((𝑎 = 𝐴𝑎 = 𝐵) ∧ (𝑏 = 𝐴𝑏 = 𝐵))) → ({𝐴, 𝐵} ∈ 𝒫 𝑉 ∧ (♯‘{𝐴, 𝐵}) = 2))
673eleq2i 2850 . . . . . . . . . . . . . . 15 ({𝐴, 𝐵} ∈ 𝑃 ↔ {𝐴, 𝐵} ∈ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2})
68 fveqeq2 6455 . . . . . . . . . . . . . . . 16 (𝑥 = {𝐴, 𝐵} → ((♯‘𝑥) = 2 ↔ (♯‘{𝐴, 𝐵}) = 2))
6968elrab 3571 . . . . . . . . . . . . . . 15 ({𝐴, 𝐵} ∈ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} ↔ ({𝐴, 𝐵} ∈ 𝒫 𝑉 ∧ (♯‘{𝐴, 𝐵}) = 2))
7067, 69bitri 267 . . . . . . . . . . . . . 14 ({𝐴, 𝐵} ∈ 𝑃 ↔ ({𝐴, 𝐵} ∈ 𝒫 𝑉 ∧ (♯‘{𝐴, 𝐵}) = 2))
7166, 70sylibr 226 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑝 = {𝑎, 𝑏})) ∧ ((𝑎 = 𝐴𝑎 = 𝐵) ∧ (𝑏 = 𝐴𝑏 = 𝐵))) → {𝐴, 𝐵} ∈ 𝑃)
72 raleq 3329 . . . . . . . . . . . . . . 15 (𝑞 = {𝐴, 𝐵} → (∀𝑥𝑞 (𝑥 = 𝐴𝑥 = 𝐵) ↔ ∀𝑥 ∈ {𝐴, 𝐵} (𝑥 = 𝐴𝑥 = 𝐵)))
73 eqeq2 2788 . . . . . . . . . . . . . . 15 (𝑞 = {𝐴, 𝐵} → ({𝑎, 𝑏} = 𝑞 ↔ {𝑎, 𝑏} = {𝐴, 𝐵}))
7472, 73imbi12d 336 . . . . . . . . . . . . . 14 (𝑞 = {𝐴, 𝐵} → ((∀𝑥𝑞 (𝑥 = 𝐴𝑥 = 𝐵) → {𝑎, 𝑏} = 𝑞) ↔ (∀𝑥 ∈ {𝐴, 𝐵} (𝑥 = 𝐴𝑥 = 𝐵) → {𝑎, 𝑏} = {𝐴, 𝐵})))
7574rspcv 3506 . . . . . . . . . . . . 13 ({𝐴, 𝐵} ∈ 𝑃 → (∀𝑞𝑃 (∀𝑥𝑞 (𝑥 = 𝐴𝑥 = 𝐵) → {𝑎, 𝑏} = 𝑞) → (∀𝑥 ∈ {𝐴, 𝐵} (𝑥 = 𝐴𝑥 = 𝐵) → {𝑎, 𝑏} = {𝐴, 𝐵})))
7671, 75syl 17 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑝 = {𝑎, 𝑏})) ∧ ((𝑎 = 𝐴𝑎 = 𝐵) ∧ (𝑏 = 𝐴𝑏 = 𝐵))) → (∀𝑞𝑃 (∀𝑥𝑞 (𝑥 = 𝐴𝑥 = 𝐵) → {𝑎, 𝑏} = 𝑞) → (∀𝑥 ∈ {𝐴, 𝐵} (𝑥 = 𝐴𝑥 = 𝐵) → {𝑎, 𝑏} = {𝐴, 𝐵})))
77 eqeq1 2781 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝐴 → (𝑥 = 𝐴𝐴 = 𝐴))
78 eqeq1 2781 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝐴 → (𝑥 = 𝐵𝐴 = 𝐵))
7977, 78orbi12d 905 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝐴 → ((𝑥 = 𝐴𝑥 = 𝐵) ↔ (𝐴 = 𝐴𝐴 = 𝐵)))
80 eqeq1 2781 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝐵 → (𝑥 = 𝐴𝐵 = 𝐴))
81 eqeq1 2781 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝐵 → (𝑥 = 𝐵𝐵 = 𝐵))
8280, 81orbi12d 905 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝐵 → ((𝑥 = 𝐴𝑥 = 𝐵) ↔ (𝐵 = 𝐴𝐵 = 𝐵)))
8379, 82ralprg 4462 . . . . . . . . . . . . . . . 16 ((𝐴𝑉𝐵𝑉) → (∀𝑥 ∈ {𝐴, 𝐵} (𝑥 = 𝐴𝑥 = 𝐵) ↔ ((𝐴 = 𝐴𝐴 = 𝐵) ∧ (𝐵 = 𝐴𝐵 = 𝐵))))
8425, 83syl 17 . . . . . . . . . . . . . . 15 (𝜑 → (∀𝑥 ∈ {𝐴, 𝐵} (𝑥 = 𝐴𝑥 = 𝐵) ↔ ((𝐴 = 𝐴𝐴 = 𝐵) ∧ (𝐵 = 𝐴𝐵 = 𝐵))))
8584imbi1d 333 . . . . . . . . . . . . . 14 (𝜑 → ((∀𝑥 ∈ {𝐴, 𝐵} (𝑥 = 𝐴𝑥 = 𝐵) → {𝑎, 𝑏} = {𝐴, 𝐵}) ↔ (((𝐴 = 𝐴𝐴 = 𝐵) ∧ (𝐵 = 𝐴𝐵 = 𝐵)) → {𝑎, 𝑏} = {𝐴, 𝐵})))
8685ad3antrrr 720 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑝 = {𝑎, 𝑏})) ∧ ((𝑎 = 𝐴𝑎 = 𝐵) ∧ (𝑏 = 𝐴𝑏 = 𝐵))) → ((∀𝑥 ∈ {𝐴, 𝐵} (𝑥 = 𝐴𝑥 = 𝐵) → {𝑎, 𝑏} = {𝐴, 𝐵}) ↔ (((𝐴 = 𝐴𝐴 = 𝐵) ∧ (𝐵 = 𝐴𝐵 = 𝐵)) → {𝑎, 𝑏} = {𝐴, 𝐵})))
87 eqid 2777 . . . . . . . . . . . . . . . 16 𝐴 = 𝐴
8887orci 854 . . . . . . . . . . . . . . 15 (𝐴 = 𝐴𝐴 = 𝐵)
89 eqid 2777 . . . . . . . . . . . . . . . 16 𝐵 = 𝐵
9089olci 855 . . . . . . . . . . . . . . 15 (𝐵 = 𝐴𝐵 = 𝐵)
91 pm5.5 353 . . . . . . . . . . . . . . 15 (((𝐴 = 𝐴𝐴 = 𝐵) ∧ (𝐵 = 𝐴𝐵 = 𝐵)) → ((((𝐴 = 𝐴𝐴 = 𝐵) ∧ (𝐵 = 𝐴𝐵 = 𝐵)) → {𝑎, 𝑏} = {𝐴, 𝐵}) ↔ {𝑎, 𝑏} = {𝐴, 𝐵}))
9288, 90, 91mp2an 682 . . . . . . . . . . . . . 14 ((((𝐴 = 𝐴𝐴 = 𝐵) ∧ (𝐵 = 𝐴𝐵 = 𝐵)) → {𝑎, 𝑏} = {𝐴, 𝐵}) ↔ {𝑎, 𝑏} = {𝐴, 𝐵})
938, 9pm3.2i 464 . . . . . . . . . . . . . . . . . . 19 (𝑎 ∈ V ∧ 𝑏 ∈ V)
94 preq12bg 4614 . . . . . . . . . . . . . . . . . . 19 (((𝑎 ∈ V ∧ 𝑏 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) → ({𝑎, 𝑏} = {𝐴, 𝐵} ↔ ((𝑎 = 𝐴𝑏 = 𝐵) ∨ (𝑎 = 𝐵𝑏 = 𝐴))))
9593, 25, 94sylancr 581 . . . . . . . . . . . . . . . . . 18 (𝜑 → ({𝑎, 𝑏} = {𝐴, 𝐵} ↔ ((𝑎 = 𝐴𝑏 = 𝐵) ∨ (𝑎 = 𝐵𝑏 = 𝐴))))
9695adantr 474 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑎𝑉𝑏𝑉)) → ({𝑎, 𝑏} = {𝐴, 𝐵} ↔ ((𝑎 = 𝐴𝑏 = 𝐵) ∨ (𝑎 = 𝐵𝑏 = 𝐴))))
9796adantr 474 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑝 = {𝑎, 𝑏})) → ({𝑎, 𝑏} = {𝐴, 𝐵} ↔ ((𝑎 = 𝐴𝑏 = 𝐵) ∨ (𝑎 = 𝐵𝑏 = 𝐴))))
98 eqeq12 2790 . . . . . . . . . . . . . . . . . . . . 21 ((𝑎 = 𝐴𝑏 = 𝐵) → (𝑎 = 𝑏𝐴 = 𝐵))
9998necon3bid 3012 . . . . . . . . . . . . . . . . . . . 20 ((𝑎 = 𝐴𝑏 = 𝐵) → (𝑎𝑏𝐴𝐵))
10099biimpd 221 . . . . . . . . . . . . . . . . . . 19 ((𝑎 = 𝐴𝑏 = 𝐵) → (𝑎𝑏𝐴𝐵))
101 eqeq12 2790 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑎 = 𝐵𝑏 = 𝐴) → (𝑎 = 𝑏𝐵 = 𝐴))
102101necon3bid 3012 . . . . . . . . . . . . . . . . . . . . 21 ((𝑎 = 𝐵𝑏 = 𝐴) → (𝑎𝑏𝐵𝐴))
103102biimpd 221 . . . . . . . . . . . . . . . . . . . 20 ((𝑎 = 𝐵𝑏 = 𝐴) → (𝑎𝑏𝐵𝐴))
104 necom 3021 . . . . . . . . . . . . . . . . . . . 20 (𝐴𝐵𝐵𝐴)
105103, 104syl6ibr 244 . . . . . . . . . . . . . . . . . . 19 ((𝑎 = 𝐵𝑏 = 𝐴) → (𝑎𝑏𝐴𝐵))
106100, 105jaoi 846 . . . . . . . . . . . . . . . . . 18 (((𝑎 = 𝐴𝑏 = 𝐵) ∨ (𝑎 = 𝐵𝑏 = 𝐴)) → (𝑎𝑏𝐴𝐵))
107106com12 32 . . . . . . . . . . . . . . . . 17 (𝑎𝑏 → (((𝑎 = 𝐴𝑏 = 𝐵) ∨ (𝑎 = 𝐵𝑏 = 𝐴)) → 𝐴𝐵))
108107ad2antrl 718 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑝 = {𝑎, 𝑏})) → (((𝑎 = 𝐴𝑏 = 𝐵) ∨ (𝑎 = 𝐵𝑏 = 𝐴)) → 𝐴𝐵))
10997, 108sylbid 232 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑝 = {𝑎, 𝑏})) → ({𝑎, 𝑏} = {𝐴, 𝐵} → 𝐴𝐵))
110109adantr 474 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑝 = {𝑎, 𝑏})) ∧ ((𝑎 = 𝐴𝑎 = 𝐵) ∧ (𝑏 = 𝐴𝑏 = 𝐵))) → ({𝑎, 𝑏} = {𝐴, 𝐵} → 𝐴𝐵))
11192, 110syl5bi 234 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑝 = {𝑎, 𝑏})) ∧ ((𝑎 = 𝐴𝑎 = 𝐵) ∧ (𝑏 = 𝐴𝑏 = 𝐵))) → ((((𝐴 = 𝐴𝐴 = 𝐵) ∧ (𝐵 = 𝐴𝐵 = 𝐵)) → {𝑎, 𝑏} = {𝐴, 𝐵}) → 𝐴𝐵))
11286, 111sylbid 232 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑝 = {𝑎, 𝑏})) ∧ ((𝑎 = 𝐴𝑎 = 𝐵) ∧ (𝑏 = 𝐴𝑏 = 𝐵))) → ((∀𝑥 ∈ {𝐴, 𝐵} (𝑥 = 𝐴𝑥 = 𝐵) → {𝑎, 𝑏} = {𝐴, 𝐵}) → 𝐴𝐵))
11376, 112syld 47 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑝 = {𝑎, 𝑏})) ∧ ((𝑎 = 𝐴𝑎 = 𝐵) ∧ (𝑏 = 𝐴𝑏 = 𝐵))) → (∀𝑞𝑃 (∀𝑥𝑞 (𝑥 = 𝐴𝑥 = 𝐵) → {𝑎, 𝑏} = 𝑞) → 𝐴𝐵))
114113ex 403 . . . . . . . . . 10 (((𝜑 ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑝 = {𝑎, 𝑏})) → (((𝑎 = 𝐴𝑎 = 𝐵) ∧ (𝑏 = 𝐴𝑏 = 𝐵)) → (∀𝑞𝑃 (∀𝑥𝑞 (𝑥 = 𝐴𝑥 = 𝐵) → {𝑎, 𝑏} = 𝑞) → 𝐴𝐵)))
115114impd 400 . . . . . . . . 9 (((𝜑 ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑝 = {𝑎, 𝑏})) → ((((𝑎 = 𝐴𝑎 = 𝐵) ∧ (𝑏 = 𝐴𝑏 = 𝐵)) ∧ ∀𝑞𝑃 (∀𝑥𝑞 (𝑥 = 𝐴𝑥 = 𝐵) → {𝑎, 𝑏} = 𝑞)) → 𝐴𝐵))
11622, 115sylbid 232 . . . . . . . 8 (((𝜑 ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑝 = {𝑎, 𝑏})) → ((∀𝑥𝑝 (𝑥 = 𝐴𝑥 = 𝐵) ∧ ∀𝑞𝑃 (∀𝑥𝑞 (𝑥 = 𝐴𝑥 = 𝐵) → 𝑝 = 𝑞)) → 𝐴𝐵))
117116ex 403 . . . . . . 7 ((𝜑 ∧ (𝑎𝑉𝑏𝑉)) → ((𝑎𝑏𝑝 = {𝑎, 𝑏}) → ((∀𝑥𝑝 (𝑥 = 𝐴𝑥 = 𝐵) ∧ ∀𝑞𝑃 (∀𝑥𝑞 (𝑥 = 𝐴𝑥 = 𝐵) → 𝑝 = 𝑞)) → 𝐴𝐵)))
118117rexlimdvva 3220 . . . . . 6 (𝜑 → (∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝑝 = {𝑎, 𝑏}) → ((∀𝑥𝑝 (𝑥 = 𝐴𝑥 = 𝐵) ∧ ∀𝑞𝑃 (∀𝑥𝑞 (𝑥 = 𝐴𝑥 = 𝐵) → 𝑝 = 𝑞)) → 𝐴𝐵)))
1196, 118syl5bi 234 . . . . 5 (𝜑 → (𝑝𝑃 → ((∀𝑥𝑝 (𝑥 = 𝐴𝑥 = 𝐵) ∧ ∀𝑞𝑃 (∀𝑥𝑞 (𝑥 = 𝐴𝑥 = 𝐵) → 𝑝 = 𝑞)) → 𝐴𝐵)))
120119imp 397 . . . 4 ((𝜑𝑝𝑃) → ((∀𝑥𝑝 (𝑥 = 𝐴𝑥 = 𝐵) ∧ ∀𝑞𝑃 (∀𝑥𝑞 (𝑥 = 𝐴𝑥 = 𝐵) → 𝑝 = 𝑞)) → 𝐴𝐵))
121120rexlimdva 3212 . . 3 (𝜑 → (∃𝑝𝑃 (∀𝑥𝑝 (𝑥 = 𝐴𝑥 = 𝐵) ∧ ∀𝑞𝑃 (∀𝑥𝑞 (𝑥 = 𝐴𝑥 = 𝐵) → 𝑝 = 𝑞)) → 𝐴𝐵))
1222, 121syl5bi 234 . 2 (𝜑 → (∃!𝑝𝑃𝑥𝑝 (𝑥 = 𝐴𝑥 = 𝐵) → 𝐴𝐵))
12327adantr 474 . . . . . . 7 ((𝜑𝐴𝐵) → {𝐴, 𝐵} ∈ 𝒫 𝑉)
124 hashprg 13497 . . . . . . . . . 10 ((𝐴𝑉𝐵𝑉) → (𝐴𝐵 ↔ (♯‘{𝐴, 𝐵}) = 2))
12525, 124syl 17 . . . . . . . . 9 (𝜑 → (𝐴𝐵 ↔ (♯‘{𝐴, 𝐵}) = 2))
126125biimpd 221 . . . . . . . 8 (𝜑 → (𝐴𝐵 → (♯‘{𝐴, 𝐵}) = 2))
127126imp 397 . . . . . . 7 ((𝜑𝐴𝐵) → (♯‘{𝐴, 𝐵}) = 2)
128123, 127jca 507 . . . . . 6 ((𝜑𝐴𝐵) → ({𝐴, 𝐵} ∈ 𝒫 𝑉 ∧ (♯‘{𝐴, 𝐵}) = 2))
129128, 70sylibr 226 . . . . 5 ((𝜑𝐴𝐵) → {𝐴, 𝐵} ∈ 𝑃)
130 raleq 3329 . . . . . . 7 (𝑝 = {𝐴, 𝐵} → (∀𝑥𝑝 (𝑥 = 𝐴𝑥 = 𝐵) ↔ ∀𝑥 ∈ {𝐴, 𝐵} (𝑥 = 𝐴𝑥 = 𝐵)))
131 eqeq1 2781 . . . . . . . . 9 (𝑝 = {𝐴, 𝐵} → (𝑝 = 𝑦 ↔ {𝐴, 𝐵} = 𝑦))
132131imbi2d 332 . . . . . . . 8 (𝑝 = {𝐴, 𝐵} → ((∀𝑥𝑦 (𝑥 = 𝐴𝑥 = 𝐵) → 𝑝 = 𝑦) ↔ (∀𝑥𝑦 (𝑥 = 𝐴𝑥 = 𝐵) → {𝐴, 𝐵} = 𝑦)))
133132ralbidv 3167 . . . . . . 7 (𝑝 = {𝐴, 𝐵} → (∀𝑦𝑃 (∀𝑥𝑦 (𝑥 = 𝐴𝑥 = 𝐵) → 𝑝 = 𝑦) ↔ ∀𝑦𝑃 (∀𝑥𝑦 (𝑥 = 𝐴𝑥 = 𝐵) → {𝐴, 𝐵} = 𝑦)))
134130, 133anbi12d 624 . . . . . 6 (𝑝 = {𝐴, 𝐵} → ((∀𝑥𝑝 (𝑥 = 𝐴𝑥 = 𝐵) ∧ ∀𝑦𝑃 (∀𝑥𝑦 (𝑥 = 𝐴𝑥 = 𝐵) → 𝑝 = 𝑦)) ↔ (∀𝑥 ∈ {𝐴, 𝐵} (𝑥 = 𝐴𝑥 = 𝐵) ∧ ∀𝑦𝑃 (∀𝑥𝑦 (𝑥 = 𝐴𝑥 = 𝐵) → {𝐴, 𝐵} = 𝑦))))
135134adantl 475 . . . . 5 (((𝜑𝐴𝐵) ∧ 𝑝 = {𝐴, 𝐵}) → ((∀𝑥𝑝 (𝑥 = 𝐴𝑥 = 𝐵) ∧ ∀𝑦𝑃 (∀𝑥𝑦 (𝑥 = 𝐴𝑥 = 𝐵) → 𝑝 = 𝑦)) ↔ (∀𝑥 ∈ {𝐴, 𝐵} (𝑥 = 𝐴𝑥 = 𝐵) ∧ ∀𝑦𝑃 (∀𝑥𝑦 (𝑥 = 𝐴𝑥 = 𝐵) → {𝐴, 𝐵} = 𝑦))))
136 vex 3400 . . . . . . . . . . 11 𝑥 ∈ V
137136elpr 4420 . . . . . . . . . 10 (𝑥 ∈ {𝐴, 𝐵} ↔ (𝑥 = 𝐴𝑥 = 𝐵))
138137a1i 11 . . . . . . . . 9 ((𝜑𝐴𝐵) → (𝑥 ∈ {𝐴, 𝐵} ↔ (𝑥 = 𝐴𝑥 = 𝐵)))
139138biimpd 221 . . . . . . . 8 ((𝜑𝐴𝐵) → (𝑥 ∈ {𝐴, 𝐵} → (𝑥 = 𝐴𝑥 = 𝐵)))
140139imp 397 . . . . . . 7 (((𝜑𝐴𝐵) ∧ 𝑥 ∈ {𝐴, 𝐵}) → (𝑥 = 𝐴𝑥 = 𝐵))
141140ralrimiva 3147 . . . . . 6 ((𝜑𝐴𝐵) → ∀𝑥 ∈ {𝐴, 𝐵} (𝑥 = 𝐴𝑥 = 𝐵))
1423eleq2i 2850 . . . . . . . . . 10 (𝑦𝑃𝑦 ∈ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2})
143 elss2prb 13583 . . . . . . . . . 10 (𝑦 ∈ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} ↔ ∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝑦 = {𝑎, 𝑏}))
144142, 143bitri 267 . . . . . . . . 9 (𝑦𝑃 ↔ ∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝑦 = {𝑎, 𝑏}))
145 prid1g 4526 . . . . . . . . . . . . . . . 16 (𝑎𝑉𝑎 ∈ {𝑎, 𝑏})
146145ad2antrl 718 . . . . . . . . . . . . . . 15 (((𝜑𝐴𝐵) ∧ (𝑎𝑉𝑏𝑉)) → 𝑎 ∈ {𝑎, 𝑏})
147146adantr 474 . . . . . . . . . . . . . 14 ((((𝜑𝐴𝐵) ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑦 = {𝑎, 𝑏})) → 𝑎 ∈ {𝑎, 𝑏})
148 eleq2 2847 . . . . . . . . . . . . . . 15 (𝑦 = {𝑎, 𝑏} → (𝑎𝑦𝑎 ∈ {𝑎, 𝑏}))
149148ad2antll 719 . . . . . . . . . . . . . 14 ((((𝜑𝐴𝐵) ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑦 = {𝑎, 𝑏})) → (𝑎𝑦𝑎 ∈ {𝑎, 𝑏}))
150147, 149mpbird 249 . . . . . . . . . . . . 13 ((((𝜑𝐴𝐵) ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑦 = {𝑎, 𝑏})) → 𝑎𝑦)
15112rspcv 3506 . . . . . . . . . . . . 13 (𝑎𝑦 → (∀𝑥𝑦 (𝑥 = 𝐴𝑥 = 𝐵) → (𝑎 = 𝐴𝑎 = 𝐵)))
152150, 151syl 17 . . . . . . . . . . . 12 ((((𝜑𝐴𝐵) ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑦 = {𝑎, 𝑏})) → (∀𝑥𝑦 (𝑥 = 𝐴𝑥 = 𝐵) → (𝑎 = 𝐴𝑎 = 𝐵)))
153 prid2g 4527 . . . . . . . . . . . . . . . . 17 (𝑏𝑉𝑏 ∈ {𝑎, 𝑏})
154153ad2antll 719 . . . . . . . . . . . . . . . 16 (((𝜑𝐴𝐵) ∧ (𝑎𝑉𝑏𝑉)) → 𝑏 ∈ {𝑎, 𝑏})
155154adantr 474 . . . . . . . . . . . . . . 15 ((((𝜑𝐴𝐵) ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑦 = {𝑎, 𝑏})) → 𝑏 ∈ {𝑎, 𝑏})
156 eleq2 2847 . . . . . . . . . . . . . . . 16 (𝑦 = {𝑎, 𝑏} → (𝑏𝑦𝑏 ∈ {𝑎, 𝑏}))
157156ad2antll 719 . . . . . . . . . . . . . . 15 ((((𝜑𝐴𝐵) ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑦 = {𝑎, 𝑏})) → (𝑏𝑦𝑏 ∈ {𝑎, 𝑏}))
158155, 157mpbird 249 . . . . . . . . . . . . . 14 ((((𝜑𝐴𝐵) ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑦 = {𝑎, 𝑏})) → 𝑏𝑦)
15915rspcv 3506 . . . . . . . . . . . . . 14 (𝑏𝑦 → (∀𝑥𝑦 (𝑥 = 𝐴𝑥 = 𝐵) → (𝑏 = 𝐴𝑏 = 𝐵)))
160158, 159syl 17 . . . . . . . . . . . . 13 ((((𝜑𝐴𝐵) ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑦 = {𝑎, 𝑏})) → (∀𝑥𝑦 (𝑥 = 𝐴𝑥 = 𝐵) → (𝑏 = 𝐴𝑏 = 𝐵)))
161 simplrr 768 . . . . . . . . . . . . . . 15 (((((𝜑𝐴𝐵) ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑦 = {𝑎, 𝑏})) ∧ ((𝑏 = 𝐴𝑏 = 𝐵) ∧ (𝑎 = 𝐴𝑎 = 𝐵))) → 𝑦 = {𝑎, 𝑏})
162 eqtr3 2800 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑎 = 𝐴𝑏 = 𝐴) → 𝑎 = 𝑏)
163 eqneqall 2979 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑎 = 𝑏 → (𝑎𝑏 → {𝑎, 𝑏} = {𝐴, 𝐵}))
164163com12 32 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑎𝑏 → (𝑎 = 𝑏 → {𝑎, 𝑏} = {𝐴, 𝐵}))
165164ad2antrl 718 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝐴𝐵) ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑦 = {𝑎, 𝑏})) → (𝑎 = 𝑏 → {𝑎, 𝑏} = {𝐴, 𝐵}))
166165com12 32 . . . . . . . . . . . . . . . . . . . . . 22 (𝑎 = 𝑏 → ((((𝜑𝐴𝐵) ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑦 = {𝑎, 𝑏})) → {𝑎, 𝑏} = {𝐴, 𝐵}))
167162, 166syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((𝑎 = 𝐴𝑏 = 𝐴) → ((((𝜑𝐴𝐵) ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑦 = {𝑎, 𝑏})) → {𝑎, 𝑏} = {𝐴, 𝐵}))
168167ex 403 . . . . . . . . . . . . . . . . . . . 20 (𝑎 = 𝐴 → (𝑏 = 𝐴 → ((((𝜑𝐴𝐵) ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑦 = {𝑎, 𝑏})) → {𝑎, 𝑏} = {𝐴, 𝐵})))
16952a1d 25 . . . . . . . . . . . . . . . . . . . . 21 ((𝑏 = 𝐴𝑎 = 𝐵) → ((((𝜑𝐴𝐵) ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑦 = {𝑎, 𝑏})) → {𝑎, 𝑏} = {𝐴, 𝐵}))
170169expcom 404 . . . . . . . . . . . . . . . . . . . 20 (𝑎 = 𝐵 → (𝑏 = 𝐴 → ((((𝜑𝐴𝐵) ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑦 = {𝑎, 𝑏})) → {𝑎, 𝑏} = {𝐴, 𝐵})))
171168, 170jaoi 846 . . . . . . . . . . . . . . . . . . 19 ((𝑎 = 𝐴𝑎 = 𝐵) → (𝑏 = 𝐴 → ((((𝜑𝐴𝐵) ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑦 = {𝑎, 𝑏})) → {𝑎, 𝑏} = {𝐴, 𝐵})))
172171com12 32 . . . . . . . . . . . . . . . . . 18 (𝑏 = 𝐴 → ((𝑎 = 𝐴𝑎 = 𝐵) → ((((𝜑𝐴𝐵) ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑦 = {𝑎, 𝑏})) → {𝑎, 𝑏} = {𝐴, 𝐵})))
17344a1d 25 . . . . . . . . . . . . . . . . . . . . 21 ((𝑎 = 𝐴𝑏 = 𝐵) → ((((𝜑𝐴𝐵) ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑦 = {𝑎, 𝑏})) → {𝑎, 𝑏} = {𝐴, 𝐵}))
174173ex 403 . . . . . . . . . . . . . . . . . . . 20 (𝑎 = 𝐴 → (𝑏 = 𝐵 → ((((𝜑𝐴𝐵) ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑦 = {𝑎, 𝑏})) → {𝑎, 𝑏} = {𝐴, 𝐵})))
175 eqtr3 2800 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑎 = 𝐵𝑏 = 𝐵) → 𝑎 = 𝑏)
176175, 166syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((𝑎 = 𝐵𝑏 = 𝐵) → ((((𝜑𝐴𝐵) ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑦 = {𝑎, 𝑏})) → {𝑎, 𝑏} = {𝐴, 𝐵}))
177176ex 403 . . . . . . . . . . . . . . . . . . . 20 (𝑎 = 𝐵 → (𝑏 = 𝐵 → ((((𝜑𝐴𝐵) ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑦 = {𝑎, 𝑏})) → {𝑎, 𝑏} = {𝐴, 𝐵})))
178174, 177jaoi 846 . . . . . . . . . . . . . . . . . . 19 ((𝑎 = 𝐴𝑎 = 𝐵) → (𝑏 = 𝐵 → ((((𝜑𝐴𝐵) ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑦 = {𝑎, 𝑏})) → {𝑎, 𝑏} = {𝐴, 𝐵})))
179178com12 32 . . . . . . . . . . . . . . . . . 18 (𝑏 = 𝐵 → ((𝑎 = 𝐴𝑎 = 𝐵) → ((((𝜑𝐴𝐵) ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑦 = {𝑎, 𝑏})) → {𝑎, 𝑏} = {𝐴, 𝐵})))
180172, 179jaoi 846 . . . . . . . . . . . . . . . . 17 ((𝑏 = 𝐴𝑏 = 𝐵) → ((𝑎 = 𝐴𝑎 = 𝐵) → ((((𝜑𝐴𝐵) ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑦 = {𝑎, 𝑏})) → {𝑎, 𝑏} = {𝐴, 𝐵})))
181180imp 397 . . . . . . . . . . . . . . . 16 (((𝑏 = 𝐴𝑏 = 𝐵) ∧ (𝑎 = 𝐴𝑎 = 𝐵)) → ((((𝜑𝐴𝐵) ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑦 = {𝑎, 𝑏})) → {𝑎, 𝑏} = {𝐴, 𝐵}))
182181impcom 398 . . . . . . . . . . . . . . 15 (((((𝜑𝐴𝐵) ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑦 = {𝑎, 𝑏})) ∧ ((𝑏 = 𝐴𝑏 = 𝐵) ∧ (𝑎 = 𝐴𝑎 = 𝐵))) → {𝑎, 𝑏} = {𝐴, 𝐵})
183161, 182eqtr2d 2814 . . . . . . . . . . . . . 14 (((((𝜑𝐴𝐵) ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑦 = {𝑎, 𝑏})) ∧ ((𝑏 = 𝐴𝑏 = 𝐵) ∧ (𝑎 = 𝐴𝑎 = 𝐵))) → {𝐴, 𝐵} = 𝑦)
184183exp32 413 . . . . . . . . . . . . 13 ((((𝜑𝐴𝐵) ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑦 = {𝑎, 𝑏})) → ((𝑏 = 𝐴𝑏 = 𝐵) → ((𝑎 = 𝐴𝑎 = 𝐵) → {𝐴, 𝐵} = 𝑦)))
185160, 184syld 47 . . . . . . . . . . . 12 ((((𝜑𝐴𝐵) ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑦 = {𝑎, 𝑏})) → (∀𝑥𝑦 (𝑥 = 𝐴𝑥 = 𝐵) → ((𝑎 = 𝐴𝑎 = 𝐵) → {𝐴, 𝐵} = 𝑦)))
186152, 185mpdd 43 . . . . . . . . . . 11 ((((𝜑𝐴𝐵) ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑦 = {𝑎, 𝑏})) → (∀𝑥𝑦 (𝑥 = 𝐴𝑥 = 𝐵) → {𝐴, 𝐵} = 𝑦))
187186ex 403 . . . . . . . . . 10 (((𝜑𝐴𝐵) ∧ (𝑎𝑉𝑏𝑉)) → ((𝑎𝑏𝑦 = {𝑎, 𝑏}) → (∀𝑥𝑦 (𝑥 = 𝐴𝑥 = 𝐵) → {𝐴, 𝐵} = 𝑦)))
188187rexlimdvva 3220 . . . . . . . . 9 ((𝜑𝐴𝐵) → (∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝑦 = {𝑎, 𝑏}) → (∀𝑥𝑦 (𝑥 = 𝐴𝑥 = 𝐵) → {𝐴, 𝐵} = 𝑦)))
189144, 188syl5bi 234 . . . . . . . 8 ((𝜑𝐴𝐵) → (𝑦𝑃 → (∀𝑥𝑦 (𝑥 = 𝐴𝑥 = 𝐵) → {𝐴, 𝐵} = 𝑦)))
190189imp 397 . . . . . . 7 (((𝜑𝐴𝐵) ∧ 𝑦𝑃) → (∀𝑥𝑦 (𝑥 = 𝐴𝑥 = 𝐵) → {𝐴, 𝐵} = 𝑦))
191190ralrimiva 3147 . . . . . 6 ((𝜑𝐴𝐵) → ∀𝑦𝑃 (∀𝑥𝑦 (𝑥 = 𝐴𝑥 = 𝐵) → {𝐴, 𝐵} = 𝑦))
192141, 191jca 507 . . . . 5 ((𝜑𝐴𝐵) → (∀𝑥 ∈ {𝐴, 𝐵} (𝑥 = 𝐴𝑥 = 𝐵) ∧ ∀𝑦𝑃 (∀𝑥𝑦 (𝑥 = 𝐴𝑥 = 𝐵) → {𝐴, 𝐵} = 𝑦)))
193129, 135, 192rspcedvd 3517 . . . 4 ((𝜑𝐴𝐵) → ∃𝑝𝑃 (∀𝑥𝑝 (𝑥 = 𝐴𝑥 = 𝐵) ∧ ∀𝑦𝑃 (∀𝑥𝑦 (𝑥 = 𝐴𝑥 = 𝐵) → 𝑝 = 𝑦)))
194 raleq 3329 . . . . 5 (𝑝 = 𝑦 → (∀𝑥𝑝 (𝑥 = 𝐴𝑥 = 𝐵) ↔ ∀𝑥𝑦 (𝑥 = 𝐴𝑥 = 𝐵)))
195194reu8 3613 . . . 4 (∃!𝑝𝑃𝑥𝑝 (𝑥 = 𝐴𝑥 = 𝐵) ↔ ∃𝑝𝑃 (∀𝑥𝑝 (𝑥 = 𝐴𝑥 = 𝐵) ∧ ∀𝑦𝑃 (∀𝑥𝑦 (𝑥 = 𝐴𝑥 = 𝐵) → 𝑝 = 𝑦)))
196193, 195sylibr 226 . . 3 ((𝜑𝐴𝐵) → ∃!𝑝𝑃𝑥𝑝 (𝑥 = 𝐴𝑥 = 𝐵))
197196ex 403 . 2 (𝜑 → (𝐴𝐵 → ∃!𝑝𝑃𝑥𝑝 (𝑥 = 𝐴𝑥 = 𝐵)))
198122, 197impbid 204 1 (𝜑 → (∃!𝑝𝑃𝑥𝑝 (𝑥 = 𝐴𝑥 = 𝐵) ↔ 𝐴𝐵))
Colors of variables: wff setvar class
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
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