Metamath Proof Explorer


Theorem dfint3

Description: Quantifier-free definition of class intersection. (Contributed by Scott Fenton, 13-Apr-2018)

Ref Expression
Assertion dfint3 𝐴 = ( V ∖ ( ( V ∖ E ) “ 𝐴 ) )

Proof

Step Hyp Ref Expression
1 dfint2 𝐴 = { 𝑥 ∣ ∀ 𝑦𝐴 𝑥𝑦 }
2 ralnex ( ∀ 𝑦𝐴 ¬ 𝑦 ( V ∖ E ) 𝑥 ↔ ¬ ∃ 𝑦𝐴 𝑦 ( V ∖ E ) 𝑥 )
3 vex 𝑦 ∈ V
4 vex 𝑥 ∈ V
5 3 4 brcnv ( 𝑦 ( V ∖ E ) 𝑥𝑥 ( V ∖ E ) 𝑦 )
6 brv 𝑥 V 𝑦
7 brdif ( 𝑥 ( V ∖ E ) 𝑦 ↔ ( 𝑥 V 𝑦 ∧ ¬ 𝑥 E 𝑦 ) )
8 6 7 mpbiran ( 𝑥 ( V ∖ E ) 𝑦 ↔ ¬ 𝑥 E 𝑦 )
9 5 8 bitr2i ( ¬ 𝑥 E 𝑦𝑦 ( V ∖ E ) 𝑥 )
10 9 con1bii ( ¬ 𝑦 ( V ∖ E ) 𝑥𝑥 E 𝑦 )
11 epel ( 𝑥 E 𝑦𝑥𝑦 )
12 10 11 bitr2i ( 𝑥𝑦 ↔ ¬ 𝑦 ( V ∖ E ) 𝑥 )
13 12 ralbii ( ∀ 𝑦𝐴 𝑥𝑦 ↔ ∀ 𝑦𝐴 ¬ 𝑦 ( V ∖ E ) 𝑥 )
14 eldif ( 𝑥 ∈ ( V ∖ ( ( V ∖ E ) “ 𝐴 ) ) ↔ ( 𝑥 ∈ V ∧ ¬ 𝑥 ∈ ( ( V ∖ E ) “ 𝐴 ) ) )
15 4 14 mpbiran ( 𝑥 ∈ ( V ∖ ( ( V ∖ E ) “ 𝐴 ) ) ↔ ¬ 𝑥 ∈ ( ( V ∖ E ) “ 𝐴 ) )
16 4 elima ( 𝑥 ∈ ( ( V ∖ E ) “ 𝐴 ) ↔ ∃ 𝑦𝐴 𝑦 ( V ∖ E ) 𝑥 )
17 15 16 xchbinx ( 𝑥 ∈ ( V ∖ ( ( V ∖ E ) “ 𝐴 ) ) ↔ ¬ ∃ 𝑦𝐴 𝑦 ( V ∖ E ) 𝑥 )
18 2 13 17 3bitr4ri ( 𝑥 ∈ ( V ∖ ( ( V ∖ E ) “ 𝐴 ) ) ↔ ∀ 𝑦𝐴 𝑥𝑦 )
19 18 abbi2i ( V ∖ ( ( V ∖ E ) “ 𝐴 ) ) = { 𝑥 ∣ ∀ 𝑦𝐴 𝑥𝑦 }
20 1 19 eqtr4i 𝐴 = ( V ∖ ( ( V ∖ E ) “ 𝐴 ) )