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Metamath Proof Explorer

Theorem pwmndid

Description: The identity of the monoid of the power set of a class A under union is the empty set. (Contributed by AV, 27-Feb-2024)

Ref Expression
Hypotheses pwmnd.b ⊢ ( Base ‘ 𝑀 ) = 𝒫 𝐴
pwmnd.p ⊢ ( +g ‘ 𝑀 ) = ( 𝑥 ∈ 𝒫 𝐴 , 𝑦 ∈ 𝒫 𝐴 ↦ ( 𝑥 ∪ 𝑦 ) )
Assertion pwmndid ( 0g ‘ 𝑀 ) = ∅

Proof

Step Hyp Ref Expression
1 pwmnd.b ⊢ ( Base ‘ 𝑀 ) = 𝒫 𝐴
2 pwmnd.p ⊢ ( +g ‘ 𝑀 ) = ( 𝑥 ∈ 𝒫 𝐴 , 𝑦 ∈ 𝒫 𝐴 ↦ ( 𝑥 ∪ 𝑦 ) )
3 0elpw ⊢ ∅ ∈ 𝒫 𝐴
4 1 eqcomi ⊢ 𝒫 𝐴 = ( Base ‘ 𝑀 )
5 eqid ⊢ ( 0g ‘ 𝑀 ) = ( 0g ‘ 𝑀 )
6 eqid ⊢ ( +g ‘ 𝑀 ) = ( +g ‘ 𝑀 )
7 id ⊢ ( ∅ ∈ 𝒫 𝐴 → ∅ ∈ 𝒫 𝐴 )
8 1 2 pwmndgplus ⊢ ( ( ∅ ∈ 𝒫 𝐴 ∧ 𝑧 ∈ 𝒫 𝐴 ) → ( ∅ ( +g ‘ 𝑀 ) 𝑧 ) = ( ∅ ∪ 𝑧 ) )
9 0un ⊢ ( ∅ ∪ 𝑧 ) = 𝑧
10 8 9 eqtrdi ⊢ ( ( ∅ ∈ 𝒫 𝐴 ∧ 𝑧 ∈ 𝒫 𝐴 ) → ( ∅ ( +g ‘ 𝑀 ) 𝑧 ) = 𝑧 )
11 1 2 pwmndgplus ⊢ ( ( 𝑧 ∈ 𝒫 𝐴 ∧ ∅ ∈ 𝒫 𝐴 ) → ( 𝑧 ( +g ‘ 𝑀 ) ∅ ) = ( 𝑧 ∪ ∅ ) )
12 11 ancoms ⊢ ( ( ∅ ∈ 𝒫 𝐴 ∧ 𝑧 ∈ 𝒫 𝐴 ) → ( 𝑧 ( +g ‘ 𝑀 ) ∅ ) = ( 𝑧 ∪ ∅ ) )
13 un0 ⊢ ( 𝑧 ∪ ∅ ) = 𝑧
14 12 13 eqtrdi ⊢ ( ( ∅ ∈ 𝒫 𝐴 ∧ 𝑧 ∈ 𝒫 𝐴 ) → ( 𝑧 ( +g ‘ 𝑀 ) ∅ ) = 𝑧 )
15 4 5 6 7 10 14 ismgmid2 ⊢ ( ∅ ∈ 𝒫 𝐴 → ∅ = ( 0g ‘ 𝑀 ) )
16 15 eqcomd ⊢ ( ∅ ∈ 𝒫 𝐴 → ( 0g ‘ 𝑀 ) = ∅ )
17 3 16 ax-mp ⊢ ( 0g ‘ 𝑀 ) = ∅