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Metamath Proof Explorer

Theorem difopab

Description: Difference of two ordered-pair class abstractions. (Contributed by Stefan O'Rear, 17-Jan-2015) (Proof shortened by SN, 19-Dec-2024)

Ref Expression
Assertion difopab ( { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } ∖ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜓 } ) = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝜑 ∧ ¬ 𝜓 ) }

Proof

Step Hyp Ref Expression
1 relopabv ⊢ Rel { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 }
2 reldif ⊢ ( Rel { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } → Rel ( { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } ∖ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜓 } ) )
3 1 2 ax-mp ⊢ Rel ( { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } ∖ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜓 } )
4 relopabv ⊢ Rel { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝜑 ∧ ¬ 𝜓 ) }
5 sban ⊢ ( [ 𝑧 / 𝑥 ] ( [ 𝑤 / 𝑦 ] 𝜑 ∧ [ 𝑤 / 𝑦 ] ¬ 𝜓 ) ↔ ( [ 𝑧 / 𝑥 ] [ 𝑤 / 𝑦 ] 𝜑 ∧ [ 𝑧 / 𝑥 ] [ 𝑤 / 𝑦 ] ¬ 𝜓 ) )
6 sban ⊢ ( [ 𝑤 / 𝑦 ] ( 𝜑 ∧ ¬ 𝜓 ) ↔ ( [ 𝑤 / 𝑦 ] 𝜑 ∧ [ 𝑤 / 𝑦 ] ¬ 𝜓 ) )
7 6 sbbii ⊢ ( [ 𝑧 / 𝑥 ] [ 𝑤 / 𝑦 ] ( 𝜑 ∧ ¬ 𝜓 ) ↔ [ 𝑧 / 𝑥 ] ( [ 𝑤 / 𝑦 ] 𝜑 ∧ [ 𝑤 / 𝑦 ] ¬ 𝜓 ) )
8 vopelopabsb ⊢ ( ⟨ 𝑧 , 𝑤 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } ↔ [ 𝑧 / 𝑥 ] [ 𝑤 / 𝑦 ] 𝜑 )
9 sbn ⊢ ( [ 𝑧 / 𝑥 ] ¬ [ 𝑤 / 𝑦 ] 𝜓 ↔ ¬ [ 𝑧 / 𝑥 ] [ 𝑤 / 𝑦 ] 𝜓 )
10 sbn ⊢ ( [ 𝑤 / 𝑦 ] ¬ 𝜓 ↔ ¬ [ 𝑤 / 𝑦 ] 𝜓 )
11 10 sbbii ⊢ ( [ 𝑧 / 𝑥 ] [ 𝑤 / 𝑦 ] ¬ 𝜓 ↔ [ 𝑧 / 𝑥 ] ¬ [ 𝑤 / 𝑦 ] 𝜓 )
12 vopelopabsb ⊢ ( ⟨ 𝑧 , 𝑤 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜓 } ↔ [ 𝑧 / 𝑥 ] [ 𝑤 / 𝑦 ] 𝜓 )
13 12 notbii ⊢ ( ¬ ⟨ 𝑧 , 𝑤 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜓 } ↔ ¬ [ 𝑧 / 𝑥 ] [ 𝑤 / 𝑦 ] 𝜓 )
14 9 11 13 3bitr4ri ⊢ ( ¬ ⟨ 𝑧 , 𝑤 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜓 } ↔ [ 𝑧 / 𝑥 ] [ 𝑤 / 𝑦 ] ¬ 𝜓 )
15 8 14 anbi12i ⊢ ( ( ⟨ 𝑧 , 𝑤 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } ∧ ¬ ⟨ 𝑧 , 𝑤 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜓 } ) ↔ ( [ 𝑧 / 𝑥 ] [ 𝑤 / 𝑦 ] 𝜑 ∧ [ 𝑧 / 𝑥 ] [ 𝑤 / 𝑦 ] ¬ 𝜓 ) )
16 5 7 15 3bitr4ri ⊢ ( ( ⟨ 𝑧 , 𝑤 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } ∧ ¬ ⟨ 𝑧 , 𝑤 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜓 } ) ↔ [ 𝑧 / 𝑥 ] [ 𝑤 / 𝑦 ] ( 𝜑 ∧ ¬ 𝜓 ) )
17 eldif ⊢ ( ⟨ 𝑧 , 𝑤 ⟩ ∈ ( { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } ∖ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜓 } ) ↔ ( ⟨ 𝑧 , 𝑤 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } ∧ ¬ ⟨ 𝑧 , 𝑤 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜓 } ) )
18 vopelopabsb ⊢ ( ⟨ 𝑧 , 𝑤 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝜑 ∧ ¬ 𝜓 ) } ↔ [ 𝑧 / 𝑥 ] [ 𝑤 / 𝑦 ] ( 𝜑 ∧ ¬ 𝜓 ) )
19 16 17 18 3bitr4i ⊢ ( ⟨ 𝑧 , 𝑤 ⟩ ∈ ( { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } ∖ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜓 } ) ↔ ⟨ 𝑧 , 𝑤 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝜑 ∧ ¬ 𝜓 ) } )
20 3 4 19 eqrelriiv ⊢ ( { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } ∖ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜓 } ) = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝜑 ∧ ¬ 𝜓 ) }