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Theorem paireqne 43658
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 3404 . . . 4 (𝑝 = 𝑞 → (∀𝑥𝑝 (𝑥 = 𝐴𝑥 = 𝐵) ↔ ∀𝑥𝑞 (𝑥 = 𝐴𝑥 = 𝐵)))
21reu8 3722 . . 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 2902 . . . . . . 7 (𝑝𝑃𝑝 ∈ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2})
5 elss2prb 13837 . . . . . . 7 (𝑝 ∈ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} ↔ ∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝑝 = {𝑎, 𝑏}))
64, 5bitri 277 . . . . . 6 (𝑝𝑃 ↔ ∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝑝 = {𝑎, 𝑏}))
7 raleq 3404 . . . . . . . . . . . 12 (𝑝 = {𝑎, 𝑏} → (∀𝑥𝑝 (𝑥 = 𝐴𝑥 = 𝐵) ↔ ∀𝑥 ∈ {𝑎, 𝑏} (𝑥 = 𝐴𝑥 = 𝐵)))
8 vex 3496 . . . . . . . . . . . . 13 𝑎 ∈ V
9 vex 3496 . . . . . . . . . . . . 13 𝑏 ∈ V
10 eqeq1 2823 . . . . . . . . . . . . . 14 (𝑥 = 𝑎 → (𝑥 = 𝐴𝑎 = 𝐴))
11 eqeq1 2823 . . . . . . . . . . . . . 14 (𝑥 = 𝑎 → (𝑥 = 𝐵𝑎 = 𝐵))
1210, 11orbi12d 914 . . . . . . . . . . . . 13 (𝑥 = 𝑎 → ((𝑥 = 𝐴𝑥 = 𝐵) ↔ (𝑎 = 𝐴𝑎 = 𝐵)))
13 eqeq1 2823 . . . . . . . . . . . . . 14 (𝑥 = 𝑏 → (𝑥 = 𝐴𝑏 = 𝐴))
14 eqeq1 2823 . . . . . . . . . . . . . 14 (𝑥 = 𝑏 → (𝑥 = 𝐵𝑏 = 𝐵))
1513, 14orbi12d 914 . . . . . . . . . . . . 13 (𝑥 = 𝑏 → ((𝑥 = 𝐴𝑥 = 𝐵) ↔ (𝑏 = 𝐴𝑏 = 𝐵)))
168, 9, 12, 15ralpr 4628 . . . . . . . . . . . 12 (∀𝑥 ∈ {𝑎, 𝑏} (𝑥 = 𝐴𝑥 = 𝐵) ↔ ((𝑎 = 𝐴𝑎 = 𝐵) ∧ (𝑏 = 𝐴𝑏 = 𝐵)))
177, 16syl6bb 289 . . . . . . . . . . 11 (𝑝 = {𝑎, 𝑏} → (∀𝑥𝑝 (𝑥 = 𝐴𝑥 = 𝐵) ↔ ((𝑎 = 𝐴𝑎 = 𝐵) ∧ (𝑏 = 𝐴𝑏 = 𝐵))))
18 eqeq1 2823 . . . . . . . . . . . . 13 (𝑝 = {𝑎, 𝑏} → (𝑝 = 𝑞 ↔ {𝑎, 𝑏} = 𝑞))
1918imbi2d 343 . . . . . . . . . . . 12 (𝑝 = {𝑎, 𝑏} → ((∀𝑥𝑞 (𝑥 = 𝐴𝑥 = 𝐵) → 𝑝 = 𝑞) ↔ (∀𝑥𝑞 (𝑥 = 𝐴𝑥 = 𝐵) → {𝑎, 𝑏} = 𝑞)))
2019ralbidv 3195 . . . . . . . . . . 11 (𝑝 = {𝑎, 𝑏} → (∀𝑞𝑃 (∀𝑥𝑞 (𝑥 = 𝐴𝑥 = 𝐵) → 𝑝 = 𝑞) ↔ ∀𝑞𝑃 (∀𝑥𝑞 (𝑥 = 𝐴𝑥 = 𝐵) → {𝑎, 𝑏} = 𝑞)))
2117, 20anbi12d 632 . . . . . . . . . 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 727 . . . . . . . . 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 514 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝐴𝑉𝐵𝑉))
26 prelpwi 5330 . . . . . . . . . . . . . . . . 17 ((𝐴𝑉𝐵𝑉) → {𝐴, 𝐵} ∈ 𝒫 𝑉)
2725, 26syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → {𝐴, 𝐵} ∈ 𝒫 𝑉)
2827ad3antrrr 728 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑝 = {𝑎, 𝑏})) ∧ ((𝑎 = 𝐴𝑎 = 𝐵) ∧ (𝑏 = 𝐴𝑏 = 𝐵))) → {𝐴, 𝐵} ∈ 𝒫 𝑉)
29 hashprg 13748 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑎𝑉𝑏𝑉) → (𝑎𝑏 ↔ (♯‘{𝑎, 𝑏}) = 2))
3029adantl 484 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑎𝑉𝑏𝑉)) → (𝑎𝑏 ↔ (♯‘{𝑎, 𝑏}) = 2))
3130biimpd 231 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑎𝑉𝑏𝑉)) → (𝑎𝑏 → (♯‘{𝑎, 𝑏}) = 2))
3231com12 32 . . . . . . . . . . . . . . . . . . 19 (𝑎𝑏 → ((𝜑 ∧ (𝑎𝑉𝑏𝑉)) → (♯‘{𝑎, 𝑏}) = 2))
3332adantr 483 . . . . . . . . . . . . . . . . . 18 ((𝑎𝑏𝑝 = {𝑎, 𝑏}) → ((𝜑 ∧ (𝑎𝑉𝑏𝑉)) → (♯‘{𝑎, 𝑏}) = 2))
3433impcom 410 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑝 = {𝑎, 𝑏})) → (♯‘{𝑎, 𝑏}) = 2)
3534adantr 483 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑝 = {𝑎, 𝑏})) ∧ ((𝑎 = 𝐴𝑎 = 𝐵) ∧ (𝑏 = 𝐴𝑏 = 𝐵))) → (♯‘{𝑎, 𝑏}) = 2)
36 eqtr3 2841 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑏 = 𝐴𝑎 = 𝐴) → 𝑏 = 𝑎)
37 eqneqall 3025 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑎 = 𝑏 → (𝑎𝑏 → (𝑝 = {𝑎, 𝑏} → {𝐴, 𝐵} = {𝑎, 𝑏})))
3837impd 413 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 413 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑎 = 𝑏 → (((𝜑 ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑝 = {𝑎, 𝑏})) → {𝐴, 𝐵} = {𝑎, 𝑏}))
4140equcoms 2020 . . . . . . . . . . . . . . . . . . . . . . . 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 415 . . . . . . . . . . . . . . . . . . . . . 22 (𝑏 = 𝐴 → (𝑎 = 𝐴 → (((𝜑 ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑝 = {𝑎, 𝑏})) → {𝐴, 𝐵} = {𝑎, 𝑏})))
44 preq12 4663 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑎 = 𝐴𝑏 = 𝐵) → {𝑎, 𝑏} = {𝐴, 𝐵})
4544eqcomd 2825 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑎 = 𝐴𝑏 = 𝐵) → {𝐴, 𝐵} = {𝑎, 𝑏})
4645a1d 25 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑎 = 𝐴𝑏 = 𝐵) → (((𝜑 ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑝 = {𝑎, 𝑏})) → {𝐴, 𝐵} = {𝑎, 𝑏}))
4746expcom 416 . . . . . . . . . . . . . . . . . . . . . 22 (𝑏 = 𝐵 → (𝑎 = 𝐴 → (((𝜑 ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑝 = {𝑎, 𝑏})) → {𝐴, 𝐵} = {𝑎, 𝑏})))
4843, 47jaoi 853 . . . . . . . . . . . . . . . . . . . . 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 4660 . . . . . . . . . . . . . . . . . . . . . . . . . 26 {𝑎, 𝑏} = {𝑏, 𝑎}
51 preq12 4663 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑏 = 𝐴𝑎 = 𝐵) → {𝑏, 𝑎} = {𝐴, 𝐵})
5250, 51syl5eq 2866 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑏 = 𝐴𝑎 = 𝐵) → {𝑎, 𝑏} = {𝐴, 𝐵})
5352eqcomd 2825 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑏 = 𝐴𝑎 = 𝐵) → {𝐴, 𝐵} = {𝑎, 𝑏})
5453a1d 25 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑏 = 𝐴𝑎 = 𝐵) → (((𝜑 ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑝 = {𝑎, 𝑏})) → {𝐴, 𝐵} = {𝑎, 𝑏}))
5554ex 415 . . . . . . . . . . . . . . . . . . . . . 22 (𝑏 = 𝐴 → (𝑎 = 𝐵 → (((𝜑 ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑝 = {𝑎, 𝑏})) → {𝐴, 𝐵} = {𝑎, 𝑏})))
56 eqtr3 2841 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑏 = 𝐵𝑎 = 𝐵) → 𝑏 = 𝑎)
5756, 41syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑏 = 𝐵𝑎 = 𝐵) → (((𝜑 ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑝 = {𝑎, 𝑏})) → {𝐴, 𝐵} = {𝑎, 𝑏}))
5857ex 415 . . . . . . . . . . . . . . . . . . . . . 22 (𝑏 = 𝐵 → (𝑎 = 𝐵 → (((𝜑 ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑝 = {𝑎, 𝑏})) → {𝐴, 𝐵} = {𝑎, 𝑏})))
5955, 58jaoi 853 . . . . . . . . . . . . . . . . . . . . 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 853 . . . . . . . . . . . . . . . . . . 19 ((𝑎 = 𝐴𝑎 = 𝐵) → ((𝑏 = 𝐴𝑏 = 𝐵) → (((𝜑 ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑝 = {𝑎, 𝑏})) → {𝐴, 𝐵} = {𝑎, 𝑏})))
6261imp 409 . . . . . . . . . . . . . . . . . 18 (((𝑎 = 𝐴𝑎 = 𝐵) ∧ (𝑏 = 𝐴𝑏 = 𝐵)) → (((𝜑 ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑝 = {𝑎, 𝑏})) → {𝐴, 𝐵} = {𝑎, 𝑏}))
6362impcom 410 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑝 = {𝑎, 𝑏})) ∧ ((𝑎 = 𝐴𝑎 = 𝐵) ∧ (𝑏 = 𝐴𝑏 = 𝐵))) → {𝐴, 𝐵} = {𝑎, 𝑏})
6463fveqeq2d 6671 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑝 = {𝑎, 𝑏})) ∧ ((𝑎 = 𝐴𝑎 = 𝐵) ∧ (𝑏 = 𝐴𝑏 = 𝐵))) → ((♯‘{𝐴, 𝐵}) = 2 ↔ (♯‘{𝑎, 𝑏}) = 2))
6535, 64mpbird 259 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑝 = {𝑎, 𝑏})) ∧ ((𝑎 = 𝐴𝑎 = 𝐵) ∧ (𝑏 = 𝐴𝑏 = 𝐵))) → (♯‘{𝐴, 𝐵}) = 2)
6628, 65jca 514 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑝 = {𝑎, 𝑏})) ∧ ((𝑎 = 𝐴𝑎 = 𝐵) ∧ (𝑏 = 𝐴𝑏 = 𝐵))) → ({𝐴, 𝐵} ∈ 𝒫 𝑉 ∧ (♯‘{𝐴, 𝐵}) = 2))
673eleq2i 2902 . . . . . . . . . . . . . . 15 ({𝐴, 𝐵} ∈ 𝑃 ↔ {𝐴, 𝐵} ∈ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2})
68 fveqeq2 6672 . . . . . . . . . . . . . . . 16 (𝑥 = {𝐴, 𝐵} → ((♯‘𝑥) = 2 ↔ (♯‘{𝐴, 𝐵}) = 2))
6968elrab 3678 . . . . . . . . . . . . . . 15 ({𝐴, 𝐵} ∈ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} ↔ ({𝐴, 𝐵} ∈ 𝒫 𝑉 ∧ (♯‘{𝐴, 𝐵}) = 2))
7067, 69bitri 277 . . . . . . . . . . . . . 14 ({𝐴, 𝐵} ∈ 𝑃 ↔ ({𝐴, 𝐵} ∈ 𝒫 𝑉 ∧ (♯‘{𝐴, 𝐵}) = 2))
7166, 70sylibr 236 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑝 = {𝑎, 𝑏})) ∧ ((𝑎 = 𝐴𝑎 = 𝐵) ∧ (𝑏 = 𝐴𝑏 = 𝐵))) → {𝐴, 𝐵} ∈ 𝑃)
72 raleq 3404 . . . . . . . . . . . . . . 15 (𝑞 = {𝐴, 𝐵} → (∀𝑥𝑞 (𝑥 = 𝐴𝑥 = 𝐵) ↔ ∀𝑥 ∈ {𝐴, 𝐵} (𝑥 = 𝐴𝑥 = 𝐵)))
73 eqeq2 2831 . . . . . . . . . . . . . . 15 (𝑞 = {𝐴, 𝐵} → ({𝑎, 𝑏} = 𝑞 ↔ {𝑎, 𝑏} = {𝐴, 𝐵}))
7472, 73imbi12d 347 . . . . . . . . . . . . . 14 (𝑞 = {𝐴, 𝐵} → ((∀𝑥𝑞 (𝑥 = 𝐴𝑥 = 𝐵) → {𝑎, 𝑏} = 𝑞) ↔ (∀𝑥 ∈ {𝐴, 𝐵} (𝑥 = 𝐴𝑥 = 𝐵) → {𝑎, 𝑏} = {𝐴, 𝐵})))
7574rspcv 3616 . . . . . . . . . . . . 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 2823 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝐴 → (𝑥 = 𝐴𝐴 = 𝐴))
78 eqeq1 2823 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝐴 → (𝑥 = 𝐵𝐴 = 𝐵))
7977, 78orbi12d 914 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝐴 → ((𝑥 = 𝐴𝑥 = 𝐵) ↔ (𝐴 = 𝐴𝐴 = 𝐵)))
80 eqeq1 2823 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝐵 → (𝑥 = 𝐴𝐵 = 𝐴))
81 eqeq1 2823 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝐵 → (𝑥 = 𝐵𝐵 = 𝐵))
8280, 81orbi12d 914 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝐵 → ((𝑥 = 𝐴𝑥 = 𝐵) ↔ (𝐵 = 𝐴𝐵 = 𝐵)))
8379, 82ralprg 4624 . . . . . . . . . . . . . . . 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 344 . . . . . . . . . . . . . 14 (𝜑 → ((∀𝑥 ∈ {𝐴, 𝐵} (𝑥 = 𝐴𝑥 = 𝐵) → {𝑎, 𝑏} = {𝐴, 𝐵}) ↔ (((𝐴 = 𝐴𝐴 = 𝐵) ∧ (𝐵 = 𝐴𝐵 = 𝐵)) → {𝑎, 𝑏} = {𝐴, 𝐵})))
8685ad3antrrr 728 . . . . . . . . . . . . 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 2819 . . . . . . . . . . . . . . . 16 𝐴 = 𝐴
8887orci 861 . . . . . . . . . . . . . . 15 (𝐴 = 𝐴𝐴 = 𝐵)
89 eqid 2819 . . . . . . . . . . . . . . . 16 𝐵 = 𝐵
9089olci 862 . . . . . . . . . . . . . . 15 (𝐵 = 𝐴𝐵 = 𝐵)
91 pm5.5 364 . . . . . . . . . . . . . . 15 (((𝐴 = 𝐴𝐴 = 𝐵) ∧ (𝐵 = 𝐴𝐵 = 𝐵)) → ((((𝐴 = 𝐴𝐴 = 𝐵) ∧ (𝐵 = 𝐴𝐵 = 𝐵)) → {𝑎, 𝑏} = {𝐴, 𝐵}) ↔ {𝑎, 𝑏} = {𝐴, 𝐵}))
9288, 90, 91mp2an 690 . . . . . . . . . . . . . 14 ((((𝐴 = 𝐴𝐴 = 𝐵) ∧ (𝐵 = 𝐴𝐵 = 𝐵)) → {𝑎, 𝑏} = {𝐴, 𝐵}) ↔ {𝑎, 𝑏} = {𝐴, 𝐵})
938, 9pm3.2i 473 . . . . . . . . . . . . . . . . . . 19 (𝑎 ∈ V ∧ 𝑏 ∈ V)
94 preq12bg 4776 . . . . . . . . . . . . . . . . . . 19 (((𝑎 ∈ V ∧ 𝑏 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) → ({𝑎, 𝑏} = {𝐴, 𝐵} ↔ ((𝑎 = 𝐴𝑏 = 𝐵) ∨ (𝑎 = 𝐵𝑏 = 𝐴))))
9593, 25, 94sylancr 589 . . . . . . . . . . . . . . . . . 18 (𝜑 → ({𝑎, 𝑏} = {𝐴, 𝐵} ↔ ((𝑎 = 𝐴𝑏 = 𝐵) ∨ (𝑎 = 𝐵𝑏 = 𝐴))))
9695adantr 483 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑎𝑉𝑏𝑉)) → ({𝑎, 𝑏} = {𝐴, 𝐵} ↔ ((𝑎 = 𝐴𝑏 = 𝐵) ∨ (𝑎 = 𝐵𝑏 = 𝐴))))
9796adantr 483 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑝 = {𝑎, 𝑏})) → ({𝑎, 𝑏} = {𝐴, 𝐵} ↔ ((𝑎 = 𝐴𝑏 = 𝐵) ∨ (𝑎 = 𝐵𝑏 = 𝐴))))
98 eqeq12 2833 . . . . . . . . . . . . . . . . . . . . 21 ((𝑎 = 𝐴𝑏 = 𝐵) → (𝑎 = 𝑏𝐴 = 𝐵))
9998necon3bid 3058 . . . . . . . . . . . . . . . . . . . 20 ((𝑎 = 𝐴𝑏 = 𝐵) → (𝑎𝑏𝐴𝐵))
10099biimpd 231 . . . . . . . . . . . . . . . . . . 19 ((𝑎 = 𝐴𝑏 = 𝐵) → (𝑎𝑏𝐴𝐵))
101 eqeq12 2833 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑎 = 𝐵𝑏 = 𝐴) → (𝑎 = 𝑏𝐵 = 𝐴))
102101necon3bid 3058 . . . . . . . . . . . . . . . . . . . . 21 ((𝑎 = 𝐵𝑏 = 𝐴) → (𝑎𝑏𝐵𝐴))
103102biimpd 231 . . . . . . . . . . . . . . . . . . . 20 ((𝑎 = 𝐵𝑏 = 𝐴) → (𝑎𝑏𝐵𝐴))
104 necom 3067 . . . . . . . . . . . . . . . . . . . 20 (𝐴𝐵𝐵𝐴)
105103, 104syl6ibr 254 . . . . . . . . . . . . . . . . . . 19 ((𝑎 = 𝐵𝑏 = 𝐴) → (𝑎𝑏𝐴𝐵))
106100, 105jaoi 853 . . . . . . . . . . . . . . . . . 18 (((𝑎 = 𝐴𝑏 = 𝐵) ∨ (𝑎 = 𝐵𝑏 = 𝐴)) → (𝑎𝑏𝐴𝐵))
107106com12 32 . . . . . . . . . . . . . . . . 17 (𝑎𝑏 → (((𝑎 = 𝐴𝑏 = 𝐵) ∨ (𝑎 = 𝐵𝑏 = 𝐴)) → 𝐴𝐵))
108107ad2antrl 726 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑝 = {𝑎, 𝑏})) → (((𝑎 = 𝐴𝑏 = 𝐵) ∨ (𝑎 = 𝐵𝑏 = 𝐴)) → 𝐴𝐵))
10997, 108sylbid 242 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑝 = {𝑎, 𝑏})) → ({𝑎, 𝑏} = {𝐴, 𝐵} → 𝐴𝐵))
110109adantr 483 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑝 = {𝑎, 𝑏})) ∧ ((𝑎 = 𝐴𝑎 = 𝐵) ∧ (𝑏 = 𝐴𝑏 = 𝐵))) → ({𝑎, 𝑏} = {𝐴, 𝐵} → 𝐴𝐵))
11192, 110syl5bi 244 . . . . . . . . . . . . 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 242 . . . . . . . . . . . 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 415 . . . . . . . . . 10 (((𝜑 ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑝 = {𝑎, 𝑏})) → (((𝑎 = 𝐴𝑎 = 𝐵) ∧ (𝑏 = 𝐴𝑏 = 𝐵)) → (∀𝑞𝑃 (∀𝑥𝑞 (𝑥 = 𝐴𝑥 = 𝐵) → {𝑎, 𝑏} = 𝑞) → 𝐴𝐵)))
115114impd 413 . . . . . . . . 9 (((𝜑 ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑝 = {𝑎, 𝑏})) → ((((𝑎 = 𝐴𝑎 = 𝐵) ∧ (𝑏 = 𝐴𝑏 = 𝐵)) ∧ ∀𝑞𝑃 (∀𝑥𝑞 (𝑥 = 𝐴𝑥 = 𝐵) → {𝑎, 𝑏} = 𝑞)) → 𝐴𝐵))
11622, 115sylbid 242 . . . . . . . 8 (((𝜑 ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑝 = {𝑎, 𝑏})) → ((∀𝑥𝑝 (𝑥 = 𝐴𝑥 = 𝐵) ∧ ∀𝑞𝑃 (∀𝑥𝑞 (𝑥 = 𝐴𝑥 = 𝐵) → 𝑝 = 𝑞)) → 𝐴𝐵))
117116ex 415 . . . . . . 7 ((𝜑 ∧ (𝑎𝑉𝑏𝑉)) → ((𝑎𝑏𝑝 = {𝑎, 𝑏}) → ((∀𝑥𝑝 (𝑥 = 𝐴𝑥 = 𝐵) ∧ ∀𝑞𝑃 (∀𝑥𝑞 (𝑥 = 𝐴𝑥 = 𝐵) → 𝑝 = 𝑞)) → 𝐴𝐵)))
118117rexlimdvva 3292 . . . . . 6 (𝜑 → (∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝑝 = {𝑎, 𝑏}) → ((∀𝑥𝑝 (𝑥 = 𝐴𝑥 = 𝐵) ∧ ∀𝑞𝑃 (∀𝑥𝑞 (𝑥 = 𝐴𝑥 = 𝐵) → 𝑝 = 𝑞)) → 𝐴𝐵)))
1196, 118syl5bi 244 . . . . 5 (𝜑 → (𝑝𝑃 → ((∀𝑥𝑝 (𝑥 = 𝐴𝑥 = 𝐵) ∧ ∀𝑞𝑃 (∀𝑥𝑞 (𝑥 = 𝐴𝑥 = 𝐵) → 𝑝 = 𝑞)) → 𝐴𝐵)))
120119imp 409 . . . 4 ((𝜑𝑝𝑃) → ((∀𝑥𝑝 (𝑥 = 𝐴𝑥 = 𝐵) ∧ ∀𝑞𝑃 (∀𝑥𝑞 (𝑥 = 𝐴𝑥 = 𝐵) → 𝑝 = 𝑞)) → 𝐴𝐵))
121120rexlimdva 3282 . . 3 (𝜑 → (∃𝑝𝑃 (∀𝑥𝑝 (𝑥 = 𝐴𝑥 = 𝐵) ∧ ∀𝑞𝑃 (∀𝑥𝑞 (𝑥 = 𝐴𝑥 = 𝐵) → 𝑝 = 𝑞)) → 𝐴𝐵))
1222, 121syl5bi 244 . 2 (𝜑 → (∃!𝑝𝑃𝑥𝑝 (𝑥 = 𝐴𝑥 = 𝐵) → 𝐴𝐵))
12327adantr 483 . . . . . . 7 ((𝜑𝐴𝐵) → {𝐴, 𝐵} ∈ 𝒫 𝑉)
124 hashprg 13748 . . . . . . . . . 10 ((𝐴𝑉𝐵𝑉) → (𝐴𝐵 ↔ (♯‘{𝐴, 𝐵}) = 2))
12525, 124syl 17 . . . . . . . . 9 (𝜑 → (𝐴𝐵 ↔ (♯‘{𝐴, 𝐵}) = 2))
126125biimpd 231 . . . . . . . 8 (𝜑 → (𝐴𝐵 → (♯‘{𝐴, 𝐵}) = 2))
127126imp 409 . . . . . . 7 ((𝜑𝐴𝐵) → (♯‘{𝐴, 𝐵}) = 2)
128123, 127jca 514 . . . . . 6 ((𝜑𝐴𝐵) → ({𝐴, 𝐵} ∈ 𝒫 𝑉 ∧ (♯‘{𝐴, 𝐵}) = 2))
129128, 70sylibr 236 . . . . 5 ((𝜑𝐴𝐵) → {𝐴, 𝐵} ∈ 𝑃)
130 raleq 3404 . . . . . . 7 (𝑝 = {𝐴, 𝐵} → (∀𝑥𝑝 (𝑥 = 𝐴𝑥 = 𝐵) ↔ ∀𝑥 ∈ {𝐴, 𝐵} (𝑥 = 𝐴𝑥 = 𝐵)))
131 eqeq1 2823 . . . . . . . . 9 (𝑝 = {𝐴, 𝐵} → (𝑝 = 𝑦 ↔ {𝐴, 𝐵} = 𝑦))
132131imbi2d 343 . . . . . . . 8 (𝑝 = {𝐴, 𝐵} → ((∀𝑥𝑦 (𝑥 = 𝐴𝑥 = 𝐵) → 𝑝 = 𝑦) ↔ (∀𝑥𝑦 (𝑥 = 𝐴𝑥 = 𝐵) → {𝐴, 𝐵} = 𝑦)))
133132ralbidv 3195 . . . . . . 7 (𝑝 = {𝐴, 𝐵} → (∀𝑦𝑃 (∀𝑥𝑦 (𝑥 = 𝐴𝑥 = 𝐵) → 𝑝 = 𝑦) ↔ ∀𝑦𝑃 (∀𝑥𝑦 (𝑥 = 𝐴𝑥 = 𝐵) → {𝐴, 𝐵} = 𝑦)))
134130, 133anbi12d 632 . . . . . 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 484 . . . . 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 3496 . . . . . . . . . . 11 𝑥 ∈ V
137136elpr 4582 . . . . . . . . . 10 (𝑥 ∈ {𝐴, 𝐵} ↔ (𝑥 = 𝐴𝑥 = 𝐵))
138137a1i 11 . . . . . . . . 9 ((𝜑𝐴𝐵) → (𝑥 ∈ {𝐴, 𝐵} ↔ (𝑥 = 𝐴𝑥 = 𝐵)))
139138biimpd 231 . . . . . . . 8 ((𝜑𝐴𝐵) → (𝑥 ∈ {𝐴, 𝐵} → (𝑥 = 𝐴𝑥 = 𝐵)))
140139imp 409 . . . . . . 7 (((𝜑𝐴𝐵) ∧ 𝑥 ∈ {𝐴, 𝐵}) → (𝑥 = 𝐴𝑥 = 𝐵))
141140ralrimiva 3180 . . . . . 6 ((𝜑𝐴𝐵) → ∀𝑥 ∈ {𝐴, 𝐵} (𝑥 = 𝐴𝑥 = 𝐵))
1423eleq2i 2902 . . . . . . . . . 10 (𝑦𝑃𝑦 ∈ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2})
143 elss2prb 13837 . . . . . . . . . 10 (𝑦 ∈ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} ↔ ∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝑦 = {𝑎, 𝑏}))
144142, 143bitri 277 . . . . . . . . 9 (𝑦𝑃 ↔ ∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝑦 = {𝑎, 𝑏}))
145 prid1g 4688 . . . . . . . . . . . . . . . 16 (𝑎𝑉𝑎 ∈ {𝑎, 𝑏})
146145ad2antrl 726 . . . . . . . . . . . . . . 15 (((𝜑𝐴𝐵) ∧ (𝑎𝑉𝑏𝑉)) → 𝑎 ∈ {𝑎, 𝑏})
147146adantr 483 . . . . . . . . . . . . . 14 ((((𝜑𝐴𝐵) ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑦 = {𝑎, 𝑏})) → 𝑎 ∈ {𝑎, 𝑏})
148 eleq2 2899 . . . . . . . . . . . . . . 15 (𝑦 = {𝑎, 𝑏} → (𝑎𝑦𝑎 ∈ {𝑎, 𝑏}))
149148ad2antll 727 . . . . . . . . . . . . . 14 ((((𝜑𝐴𝐵) ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑦 = {𝑎, 𝑏})) → (𝑎𝑦𝑎 ∈ {𝑎, 𝑏}))
150147, 149mpbird 259 . . . . . . . . . . . . 13 ((((𝜑𝐴𝐵) ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑦 = {𝑎, 𝑏})) → 𝑎𝑦)
15112rspcv 3616 . . . . . . . . . . . . 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 4689 . . . . . . . . . . . . . . . . 17 (𝑏𝑉𝑏 ∈ {𝑎, 𝑏})
154153ad2antll 727 . . . . . . . . . . . . . . . 16 (((𝜑𝐴𝐵) ∧ (𝑎𝑉𝑏𝑉)) → 𝑏 ∈ {𝑎, 𝑏})
155154adantr 483 . . . . . . . . . . . . . . 15 ((((𝜑𝐴𝐵) ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑦 = {𝑎, 𝑏})) → 𝑏 ∈ {𝑎, 𝑏})
156 eleq2 2899 . . . . . . . . . . . . . . . 16 (𝑦 = {𝑎, 𝑏} → (𝑏𝑦𝑏 ∈ {𝑎, 𝑏}))
157156ad2antll 727 . . . . . . . . . . . . . . 15 ((((𝜑𝐴𝐵) ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑦 = {𝑎, 𝑏})) → (𝑏𝑦𝑏 ∈ {𝑎, 𝑏}))
158155, 157mpbird 259 . . . . . . . . . . . . . 14 ((((𝜑𝐴𝐵) ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑦 = {𝑎, 𝑏})) → 𝑏𝑦)
15915rspcv 3616 . . . . . . . . . . . . . 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 776 . . . . . . . . . . . . . . 15 (((((𝜑𝐴𝐵) ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑦 = {𝑎, 𝑏})) ∧ ((𝑏 = 𝐴𝑏 = 𝐵) ∧ (𝑎 = 𝐴𝑎 = 𝐵))) → 𝑦 = {𝑎, 𝑏})
162 eqtr3 2841 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑎 = 𝐴𝑏 = 𝐴) → 𝑎 = 𝑏)
163 eqneqall 3025 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑎 = 𝑏 → (𝑎𝑏 → {𝑎, 𝑏} = {𝐴, 𝐵}))
164163com12 32 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑎𝑏 → (𝑎 = 𝑏 → {𝑎, 𝑏} = {𝐴, 𝐵}))
165164ad2antrl 726 . . . . . . . . . . . . . . . . . . . . . . 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 415 . . . . . . . . . . . . . . . . . . . 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 416 . . . . . . . . . . . . . . . . . . . 20 (𝑎 = 𝐵 → (𝑏 = 𝐴 → ((((𝜑𝐴𝐵) ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑦 = {𝑎, 𝑏})) → {𝑎, 𝑏} = {𝐴, 𝐵})))
171168, 170jaoi 853 . . . . . . . . . . . . . . . . . . 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 415 . . . . . . . . . . . . . . . . . . . 20 (𝑎 = 𝐴 → (𝑏 = 𝐵 → ((((𝜑𝐴𝐵) ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑦 = {𝑎, 𝑏})) → {𝑎, 𝑏} = {𝐴, 𝐵})))
175 eqtr3 2841 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑎 = 𝐵𝑏 = 𝐵) → 𝑎 = 𝑏)
176175, 166syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((𝑎 = 𝐵𝑏 = 𝐵) → ((((𝜑𝐴𝐵) ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑦 = {𝑎, 𝑏})) → {𝑎, 𝑏} = {𝐴, 𝐵}))
177176ex 415 . . . . . . . . . . . . . . . . . . . 20 (𝑎 = 𝐵 → (𝑏 = 𝐵 → ((((𝜑𝐴𝐵) ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑦 = {𝑎, 𝑏})) → {𝑎, 𝑏} = {𝐴, 𝐵})))
178174, 177jaoi 853 . . . . . . . . . . . . . . . . . . 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 853 . . . . . . . . . . . . . . . . 17 ((𝑏 = 𝐴𝑏 = 𝐵) → ((𝑎 = 𝐴𝑎 = 𝐵) → ((((𝜑𝐴𝐵) ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑦 = {𝑎, 𝑏})) → {𝑎, 𝑏} = {𝐴, 𝐵})))
181180imp 409 . . . . . . . . . . . . . . . 16 (((𝑏 = 𝐴𝑏 = 𝐵) ∧ (𝑎 = 𝐴𝑎 = 𝐵)) → ((((𝜑𝐴𝐵) ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑦 = {𝑎, 𝑏})) → {𝑎, 𝑏} = {𝐴, 𝐵}))
182181impcom 410 . . . . . . . . . . . . . . 15 (((((𝜑𝐴𝐵) ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑦 = {𝑎, 𝑏})) ∧ ((𝑏 = 𝐴𝑏 = 𝐵) ∧ (𝑎 = 𝐴𝑎 = 𝐵))) → {𝑎, 𝑏} = {𝐴, 𝐵})
183161, 182eqtr2d 2855 . . . . . . . . . . . . . 14 (((((𝜑𝐴𝐵) ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑦 = {𝑎, 𝑏})) ∧ ((𝑏 = 𝐴𝑏 = 𝐵) ∧ (𝑎 = 𝐴𝑎 = 𝐵))) → {𝐴, 𝐵} = 𝑦)
184183exp32 423 . . . . . . . . . . . . 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 415 . . . . . . . . . 10 (((𝜑𝐴𝐵) ∧ (𝑎𝑉𝑏𝑉)) → ((𝑎𝑏𝑦 = {𝑎, 𝑏}) → (∀𝑥𝑦 (𝑥 = 𝐴𝑥 = 𝐵) → {𝐴, 𝐵} = 𝑦)))
188187rexlimdvva 3292 . . . . . . . . 9 ((𝜑𝐴𝐵) → (∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝑦 = {𝑎, 𝑏}) → (∀𝑥𝑦 (𝑥 = 𝐴𝑥 = 𝐵) → {𝐴, 𝐵} = 𝑦)))
189144, 188syl5bi 244 . . . . . . . 8 ((𝜑𝐴𝐵) → (𝑦𝑃 → (∀𝑥𝑦 (𝑥 = 𝐴𝑥 = 𝐵) → {𝐴, 𝐵} = 𝑦)))
190189imp 409 . . . . . . 7 (((𝜑𝐴𝐵) ∧ 𝑦𝑃) → (∀𝑥𝑦 (𝑥 = 𝐴𝑥 = 𝐵) → {𝐴, 𝐵} = 𝑦))
191190ralrimiva 3180 . . . . . 6 ((𝜑𝐴𝐵) → ∀𝑦𝑃 (∀𝑥𝑦 (𝑥 = 𝐴𝑥 = 𝐵) → {𝐴, 𝐵} = 𝑦))
192141, 191jca 514 . . . . 5 ((𝜑𝐴𝐵) → (∀𝑥 ∈ {𝐴, 𝐵} (𝑥 = 𝐴𝑥 = 𝐵) ∧ ∀𝑦𝑃 (∀𝑥𝑦 (𝑥 = 𝐴𝑥 = 𝐵) → {𝐴, 𝐵} = 𝑦)))
193129, 135, 192rspcedvd 3624 . . . 4 ((𝜑𝐴𝐵) → ∃𝑝𝑃 (∀𝑥𝑝 (𝑥 = 𝐴𝑥 = 𝐵) ∧ ∀𝑦𝑃 (∀𝑥𝑦 (𝑥 = 𝐴𝑥 = 𝐵) → 𝑝 = 𝑦)))
194 raleq 3404 . . . . 5 (𝑝 = 𝑦 → (∀𝑥𝑝 (𝑥 = 𝐴𝑥 = 𝐵) ↔ ∀𝑥𝑦 (𝑥 = 𝐴𝑥 = 𝐵)))
195194reu8 3722 . . . 4 (∃!𝑝𝑃𝑥𝑝 (𝑥 = 𝐴𝑥 = 𝐵) ↔ ∃𝑝𝑃 (∀𝑥𝑝 (𝑥 = 𝐴𝑥 = 𝐵) ∧ ∀𝑦𝑃 (∀𝑥𝑦 (𝑥 = 𝐴𝑥 = 𝐵) → 𝑝 = 𝑦)))
196193, 195sylibr 236 . . 3 ((𝜑𝐴𝐵) → ∃!𝑝𝑃𝑥𝑝 (𝑥 = 𝐴𝑥 = 𝐵))
197196ex 415 . 2 (𝜑 → (𝐴𝐵 → ∃!𝑝𝑃𝑥𝑝 (𝑥 = 𝐴𝑥 = 𝐵)))
198122, 197impbid 214 1 (𝜑 → (∃!𝑝𝑃𝑥𝑝 (𝑥 = 𝐴𝑥 = 𝐵) ↔ 𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  wo 843   = wceq 1530  wcel 2107  wne 3014  wral 3136  wrex 3137  ∃!wreu 3138  {crab 3140  Vcvv 3493  𝒫 cpw 4537  {cpr 4561  cfv 6348  2c2 11684  chash 13682
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-nel 3122  df-ral 3141  df-rex 3142  df-reu 3143  df-rmo 3144  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7573  df-1st 7681  df-2nd 7682  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-1o 8094  df-2o 8095  df-oadd 8098  df-er 8281  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-dju 9322  df-card 9360  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11631  df-2 11692  df-n0 11890  df-z 11974  df-uz 12236  df-fz 12885  df-hash 13683
This theorem is referenced by:  requad2  43773
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