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Metamath Proof Explorer

Theorem hashle2prv

Description: A nonempty subset of a powerset of a class V has size less than or equal to two iff it is an unordered pair of elements of V . (Contributed by AV, 24-Nov-2021)

Ref Expression
Assertion hashle2prv ( 𝑃 ∈ ( 𝒫 𝑉 ∖ { ∅ } ) → ( ( ♯ ‘ 𝑃 ) ≤ 2 ↔ ∃ 𝑎 ∈ 𝑉 ∃ 𝑏 ∈ 𝑉 𝑃 = { 𝑎 , 𝑏 } ) )

Proof

Step Hyp Ref Expression
1 eldifsn ⊢ ( 𝑃 ∈ ( 𝒫 𝑉 ∖ { ∅ } ) ↔ ( 𝑃 ∈ 𝒫 𝑉 ∧ 𝑃 ≠ ∅ ) )
2 hashle2pr ⊢ ( ( 𝑃 ∈ 𝒫 𝑉 ∧ 𝑃 ≠ ∅ ) → ( ( ♯ ‘ 𝑃 ) ≤ 2 ↔ ∃ 𝑎 ∃ 𝑏 𝑃 = { 𝑎 , 𝑏 } ) )
3 1 2 sylbi ⊢ ( 𝑃 ∈ ( 𝒫 𝑉 ∖ { ∅ } ) → ( ( ♯ ‘ 𝑃 ) ≤ 2 ↔ ∃ 𝑎 ∃ 𝑏 𝑃 = { 𝑎 , 𝑏 } ) )
4 eldifi ⊢ ( 𝑃 ∈ ( 𝒫 𝑉 ∖ { ∅ } ) → 𝑃 ∈ 𝒫 𝑉 )
5 eleq1 ⊢ ( 𝑃 = { 𝑎 , 𝑏 } → ( 𝑃 ∈ 𝒫 𝑉 ↔ { 𝑎 , 𝑏 } ∈ 𝒫 𝑉 ) )
6 prelpw ⊢ ( ( 𝑎 ∈ V ∧ 𝑏 ∈ V ) → ( ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ↔ { 𝑎 , 𝑏 } ∈ 𝒫 𝑉 ) )
7 6 biimprd ⊢ ( ( 𝑎 ∈ V ∧ 𝑏 ∈ V ) → ( { 𝑎 , 𝑏 } ∈ 𝒫 𝑉 → ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ) )
8 7 el2v ⊢ ( { 𝑎 , 𝑏 } ∈ 𝒫 𝑉 → ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) )
9 5 8 biimtrdi ⊢ ( 𝑃 = { 𝑎 , 𝑏 } → ( 𝑃 ∈ 𝒫 𝑉 → ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ) )
10 4 9 syl5com ⊢ ( 𝑃 ∈ ( 𝒫 𝑉 ∖ { ∅ } ) → ( 𝑃 = { 𝑎 , 𝑏 } → ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ) )
11 10 pm4.71rd ⊢ ( 𝑃 ∈ ( 𝒫 𝑉 ∖ { ∅ } ) → ( 𝑃 = { 𝑎 , 𝑏 } ↔ ( ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ∧ 𝑃 = { 𝑎 , 𝑏 } ) ) )
12 11 2exbidv ⊢ ( 𝑃 ∈ ( 𝒫 𝑉 ∖ { ∅ } ) → ( ∃ 𝑎 ∃ 𝑏 𝑃 = { 𝑎 , 𝑏 } ↔ ∃ 𝑎 ∃ 𝑏 ( ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ∧ 𝑃 = { 𝑎 , 𝑏 } ) ) )
13 r2ex ⊢ ( ∃ 𝑎 ∈ 𝑉 ∃ 𝑏 ∈ 𝑉 𝑃 = { 𝑎 , 𝑏 } ↔ ∃ 𝑎 ∃ 𝑏 ( ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ∧ 𝑃 = { 𝑎 , 𝑏 } ) )
14 13 bicomi ⊢ ( ∃ 𝑎 ∃ 𝑏 ( ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ∧ 𝑃 = { 𝑎 , 𝑏 } ) ↔ ∃ 𝑎 ∈ 𝑉 ∃ 𝑏 ∈ 𝑉 𝑃 = { 𝑎 , 𝑏 } )
15 14 a1i ⊢ ( 𝑃 ∈ ( 𝒫 𝑉 ∖ { ∅ } ) → ( ∃ 𝑎 ∃ 𝑏 ( ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ∧ 𝑃 = { 𝑎 , 𝑏 } ) ↔ ∃ 𝑎 ∈ 𝑉 ∃ 𝑏 ∈ 𝑉 𝑃 = { 𝑎 , 𝑏 } ) )
16 3 12 15 3bitrd ⊢ ( 𝑃 ∈ ( 𝒫 𝑉 ∖ { ∅ } ) → ( ( ♯ ‘ 𝑃 ) ≤ 2 ↔ ∃ 𝑎 ∈ 𝑉 ∃ 𝑏 ∈ 𝑉 𝑃 = { 𝑎 , 𝑏 } ) )