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Metamath Proof Explorer

Theorem hash1elsn

Description: A set of size 1 with a known element is the singleton of that element. (Contributed by Rohan Ridenour, 3-Aug-2023)

Ref Expression
Hypotheses hash1elsn.1 ⊢ ( 𝜑 → ( ♯ ‘ 𝐴 ) = 1 )
hash1elsn.2 ⊢ ( 𝜑 → 𝐵 ∈ 𝐴 )
hash1elsn.3 ⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
Assertion hash1elsn ( 𝜑 → 𝐴 = { 𝐵 } )

Proof

Step Hyp Ref Expression
1 hash1elsn.1 ⊢ ( 𝜑 → ( ♯ ‘ 𝐴 ) = 1 )
2 hash1elsn.2 ⊢ ( 𝜑 → 𝐵 ∈ 𝐴 )
3 hash1elsn.3 ⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
4 hashen1 ⊢ ( 𝐴 ∈ 𝑉 → ( ( ♯ ‘ 𝐴 ) = 1 ↔ 𝐴 ≈ 1o ) )
5 3 4 syl ⊢ ( 𝜑 → ( ( ♯ ‘ 𝐴 ) = 1 ↔ 𝐴 ≈ 1o ) )
6 1 5 mpbid ⊢ ( 𝜑 → 𝐴 ≈ 1o )
7 en1 ⊢ ( 𝐴 ≈ 1o ↔ ∃ 𝑥 𝐴 = { 𝑥 } )
8 6 7 sylib ⊢ ( 𝜑 → ∃ 𝑥 𝐴 = { 𝑥 } )
9 simpr ⊢ ( ( 𝜑 ∧ 𝐴 = { 𝑥 } ) → 𝐴 = { 𝑥 } )
10 2 adantr ⊢ ( ( 𝜑 ∧ 𝐴 = { 𝑥 } ) → 𝐵 ∈ 𝐴 )
11 10 9 eleqtrd ⊢ ( ( 𝜑 ∧ 𝐴 = { 𝑥 } ) → 𝐵 ∈ { 𝑥 } )
12 elsni ⊢ ( 𝐵 ∈ { 𝑥 } → 𝐵 = 𝑥 )
13 11 12 syl ⊢ ( ( 𝜑 ∧ 𝐴 = { 𝑥 } ) → 𝐵 = 𝑥 )
14 13 sneqd ⊢ ( ( 𝜑 ∧ 𝐴 = { 𝑥 } ) → { 𝐵 } = { 𝑥 } )
15 9 14 eqtr4d ⊢ ( ( 𝜑 ∧ 𝐴 = { 𝑥 } ) → 𝐴 = { 𝐵 } )
16 8 15 exlimddv ⊢ ( 𝜑 → 𝐴 = { 𝐵 } )