# Metamath Proof Explorer

## Theorem meet0

Description: Lemma for odujoin . (Contributed by Stefan O'Rear, 29-Jan-2015) TODO ( df-riota update): This proof increased from 152 bytes to 547 bytes after the df-riota change. Any way to shorten it? join0 also.

Ref Expression
Assertion meet0 ( meet ‘ ∅ ) = ∅

### Proof

Step Hyp Ref Expression
1 0ex ∅ ∈ V
2 eqid ( glb ‘ ∅ ) = ( glb ‘ ∅ )
3 eqid ( meet ‘ ∅ ) = ( meet ‘ ∅ )
4 2 3 meetfval ( ∅ ∈ V → ( meet ‘ ∅ ) = { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ { 𝑥 , 𝑦 } ( glb ‘ ∅ ) 𝑧 } )
5 1 4 ax-mp ( meet ‘ ∅ ) = { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ { 𝑥 , 𝑦 } ( glb ‘ ∅ ) 𝑧 }
6 df-oprab { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ { 𝑥 , 𝑦 } ( glb ‘ ∅ ) 𝑧 } = { 𝑤 ∣ ∃ 𝑥𝑦𝑧 ( 𝑤 = ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∧ { 𝑥 , 𝑦 } ( glb ‘ ∅ ) 𝑧 ) }
7 br0 ¬ { 𝑥 , 𝑦 } ∅ 𝑧
8 base0 ∅ = ( Base ‘ ∅ )
9 eqid ( le ‘ ∅ ) = ( le ‘ ∅ )
10 biid ( ( ∀ 𝑧𝑥 𝑦 ( le ‘ ∅ ) 𝑧 ∧ ∀ 𝑤 ∈ ∅ ( ∀ 𝑧𝑥 𝑤 ( le ‘ ∅ ) 𝑧𝑤 ( le ‘ ∅ ) 𝑦 ) ) ↔ ( ∀ 𝑧𝑥 𝑦 ( le ‘ ∅ ) 𝑧 ∧ ∀ 𝑤 ∈ ∅ ( ∀ 𝑧𝑥 𝑤 ( le ‘ ∅ ) 𝑧𝑤 ( le ‘ ∅ ) 𝑦 ) ) )
11 id ( ∅ ∈ V → ∅ ∈ V )
12 8 9 2 10 11 glbfval ( ∅ ∈ V → ( glb ‘ ∅ ) = ( ( 𝑥 ∈ 𝒫 ∅ ↦ ( 𝑦 ∈ ∅ ( ∀ 𝑧𝑥 𝑦 ( le ‘ ∅ ) 𝑧 ∧ ∀ 𝑤 ∈ ∅ ( ∀ 𝑧𝑥 𝑤 ( le ‘ ∅ ) 𝑧𝑤 ( le ‘ ∅ ) 𝑦 ) ) ) ) ↾ { 𝑥 ∣ ∃! 𝑦 ∈ ∅ ( ∀ 𝑧𝑥 𝑦 ( le ‘ ∅ ) 𝑧 ∧ ∀ 𝑤 ∈ ∅ ( ∀ 𝑧𝑥 𝑤 ( le ‘ ∅ ) 𝑧𝑤 ( le ‘ ∅ ) 𝑦 ) ) } ) )
13 1 12 ax-mp ( glb ‘ ∅ ) = ( ( 𝑥 ∈ 𝒫 ∅ ↦ ( 𝑦 ∈ ∅ ( ∀ 𝑧𝑥 𝑦 ( le ‘ ∅ ) 𝑧 ∧ ∀ 𝑤 ∈ ∅ ( ∀ 𝑧𝑥 𝑤 ( le ‘ ∅ ) 𝑧𝑤 ( le ‘ ∅ ) 𝑦 ) ) ) ) ↾ { 𝑥 ∣ ∃! 𝑦 ∈ ∅ ( ∀ 𝑧𝑥 𝑦 ( le ‘ ∅ ) 𝑧 ∧ ∀ 𝑤 ∈ ∅ ( ∀ 𝑧𝑥 𝑤 ( le ‘ ∅ ) 𝑧𝑤 ( le ‘ ∅ ) 𝑦 ) ) } )
14 reu0 ¬ ∃! 𝑦 ∈ ∅ ( ∀ 𝑧𝑥 𝑦 ( le ‘ ∅ ) 𝑧 ∧ ∀ 𝑤 ∈ ∅ ( ∀ 𝑧𝑥 𝑤 ( le ‘ ∅ ) 𝑧𝑤 ( le ‘ ∅ ) 𝑦 ) )
15 14 abf { 𝑥 ∣ ∃! 𝑦 ∈ ∅ ( ∀ 𝑧𝑥 𝑦 ( le ‘ ∅ ) 𝑧 ∧ ∀ 𝑤 ∈ ∅ ( ∀ 𝑧𝑥 𝑤 ( le ‘ ∅ ) 𝑧𝑤 ( le ‘ ∅ ) 𝑦 ) ) } = ∅
16 15 reseq2i ( ( 𝑥 ∈ 𝒫 ∅ ↦ ( 𝑦 ∈ ∅ ( ∀ 𝑧𝑥 𝑦 ( le ‘ ∅ ) 𝑧 ∧ ∀ 𝑤 ∈ ∅ ( ∀ 𝑧𝑥 𝑤 ( le ‘ ∅ ) 𝑧𝑤 ( le ‘ ∅ ) 𝑦 ) ) ) ) ↾ { 𝑥 ∣ ∃! 𝑦 ∈ ∅ ( ∀ 𝑧𝑥 𝑦 ( le ‘ ∅ ) 𝑧 ∧ ∀ 𝑤 ∈ ∅ ( ∀ 𝑧𝑥 𝑤 ( le ‘ ∅ ) 𝑧𝑤 ( le ‘ ∅ ) 𝑦 ) ) } ) = ( ( 𝑥 ∈ 𝒫 ∅ ↦ ( 𝑦 ∈ ∅ ( ∀ 𝑧𝑥 𝑦 ( le ‘ ∅ ) 𝑧 ∧ ∀ 𝑤 ∈ ∅ ( ∀ 𝑧𝑥 𝑤 ( le ‘ ∅ ) 𝑧𝑤 ( le ‘ ∅ ) 𝑦 ) ) ) ) ↾ ∅ )
17 res0 ( ( 𝑥 ∈ 𝒫 ∅ ↦ ( 𝑦 ∈ ∅ ( ∀ 𝑧𝑥 𝑦 ( le ‘ ∅ ) 𝑧 ∧ ∀ 𝑤 ∈ ∅ ( ∀ 𝑧𝑥 𝑤 ( le ‘ ∅ ) 𝑧𝑤 ( le ‘ ∅ ) 𝑦 ) ) ) ) ↾ ∅ ) = ∅
18 16 17 eqtri ( ( 𝑥 ∈ 𝒫 ∅ ↦ ( 𝑦 ∈ ∅ ( ∀ 𝑧𝑥 𝑦 ( le ‘ ∅ ) 𝑧 ∧ ∀ 𝑤 ∈ ∅ ( ∀ 𝑧𝑥 𝑤 ( le ‘ ∅ ) 𝑧𝑤 ( le ‘ ∅ ) 𝑦 ) ) ) ) ↾ { 𝑥 ∣ ∃! 𝑦 ∈ ∅ ( ∀ 𝑧𝑥 𝑦 ( le ‘ ∅ ) 𝑧 ∧ ∀ 𝑤 ∈ ∅ ( ∀ 𝑧𝑥 𝑤 ( le ‘ ∅ ) 𝑧𝑤 ( le ‘ ∅ ) 𝑦 ) ) } ) = ∅
19 13 18 eqtri ( glb ‘ ∅ ) = ∅
20 19 breqi ( { 𝑥 , 𝑦 } ( glb ‘ ∅ ) 𝑧 ↔ { 𝑥 , 𝑦 } ∅ 𝑧 )
21 7 20 mtbir ¬ { 𝑥 , 𝑦 } ( glb ‘ ∅ ) 𝑧
22 21 intnan ¬ ( 𝑤 = ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∧ { 𝑥 , 𝑦 } ( glb ‘ ∅ ) 𝑧 )
23 22 nex ¬ ∃ 𝑧 ( 𝑤 = ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∧ { 𝑥 , 𝑦 } ( glb ‘ ∅ ) 𝑧 )
24 23 nex ¬ ∃ 𝑦𝑧 ( 𝑤 = ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∧ { 𝑥 , 𝑦 } ( glb ‘ ∅ ) 𝑧 )
25 24 nex ¬ ∃ 𝑥𝑦𝑧 ( 𝑤 = ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∧ { 𝑥 , 𝑦 } ( glb ‘ ∅ ) 𝑧 )
26 25 abf { 𝑤 ∣ ∃ 𝑥𝑦𝑧 ( 𝑤 = ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∧ { 𝑥 , 𝑦 } ( glb ‘ ∅ ) 𝑧 ) } = ∅
27 6 26 eqtri { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ { 𝑥 , 𝑦 } ( glb ‘ ∅ ) 𝑧 } = ∅
28 5 27 eqtri ( meet ‘ ∅ ) = ∅