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 ↔ ∃ 𝑎𝑉𝑏𝑉 𝑃 = { 𝑎 , 𝑏 } ) )

Proof

Step Hyp Ref Expression
1 eldifsn ( 𝑃 ∈ ( 𝒫 𝑉 ∖ { ∅ } ) ↔ ( 𝑃 ∈ 𝒫 𝑉𝑃 ≠ ∅ ) )
2 hashle2pr ( ( 𝑃 ∈ 𝒫 𝑉𝑃 ≠ ∅ ) → ( ( ♯ ‘ 𝑃 ) ≤ 2 ↔ ∃ 𝑎𝑏 𝑃 = { 𝑎 , 𝑏 } ) )
3 1 2 sylbi ( 𝑃 ∈ ( 𝒫 𝑉 ∖ { ∅ } ) → ( ( ♯ ‘ 𝑃 ) ≤ 2 ↔ ∃ 𝑎𝑏 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 ) → ( ( 𝑎𝑉𝑏𝑉 ) ↔ { 𝑎 , 𝑏 } ∈ 𝒫 𝑉 ) )
7 6 biimprd ( ( 𝑎 ∈ V ∧ 𝑏 ∈ V ) → ( { 𝑎 , 𝑏 } ∈ 𝒫 𝑉 → ( 𝑎𝑉𝑏𝑉 ) ) )
8 7 el2v ( { 𝑎 , 𝑏 } ∈ 𝒫 𝑉 → ( 𝑎𝑉𝑏𝑉 ) )
9 5 8 syl6bi ( 𝑃 = { 𝑎 , 𝑏 } → ( 𝑃 ∈ 𝒫 𝑉 → ( 𝑎𝑉𝑏𝑉 ) ) )
10 4 9 syl5com ( 𝑃 ∈ ( 𝒫 𝑉 ∖ { ∅ } ) → ( 𝑃 = { 𝑎 , 𝑏 } → ( 𝑎𝑉𝑏𝑉 ) ) )
11 10 pm4.71rd ( 𝑃 ∈ ( 𝒫 𝑉 ∖ { ∅ } ) → ( 𝑃 = { 𝑎 , 𝑏 } ↔ ( ( 𝑎𝑉𝑏𝑉 ) ∧ 𝑃 = { 𝑎 , 𝑏 } ) ) )
12 11 2exbidv ( 𝑃 ∈ ( 𝒫 𝑉 ∖ { ∅ } ) → ( ∃ 𝑎𝑏 𝑃 = { 𝑎 , 𝑏 } ↔ ∃ 𝑎𝑏 ( ( 𝑎𝑉𝑏𝑉 ) ∧ 𝑃 = { 𝑎 , 𝑏 } ) ) )
13 r2ex ( ∃ 𝑎𝑉𝑏𝑉 𝑃 = { 𝑎 , 𝑏 } ↔ ∃ 𝑎𝑏 ( ( 𝑎𝑉𝑏𝑉 ) ∧ 𝑃 = { 𝑎 , 𝑏 } ) )
14 13 bicomi ( ∃ 𝑎𝑏 ( ( 𝑎𝑉𝑏𝑉 ) ∧ 𝑃 = { 𝑎 , 𝑏 } ) ↔ ∃ 𝑎𝑉𝑏𝑉 𝑃 = { 𝑎 , 𝑏 } )
15 14 a1i ( 𝑃 ∈ ( 𝒫 𝑉 ∖ { ∅ } ) → ( ∃ 𝑎𝑏 ( ( 𝑎𝑉𝑏𝑉 ) ∧ 𝑃 = { 𝑎 , 𝑏 } ) ↔ ∃ 𝑎𝑉𝑏𝑉 𝑃 = { 𝑎 , 𝑏 } ) )
16 3 12 15 3bitrd ( 𝑃 ∈ ( 𝒫 𝑉 ∖ { ∅ } ) → ( ( ♯ ‘ 𝑃 ) ≤ 2 ↔ ∃ 𝑎𝑉𝑏𝑉 𝑃 = { 𝑎 , 𝑏 } ) )