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Metamath Proof Explorer

Theorem undif3

Description: An equality involving class union and class difference. The first equality of Exercise 13 of TakeutiZaring p. 22. (Contributed by Alan Sare, 17-Apr-2012) (Proof shortened by JJ, 13-Jul-2021)

Ref Expression
Assertion undif3 ( 𝐴 ∪ ( 𝐵 ∖ 𝐶 ) ) = ( ( 𝐴 ∪ 𝐵 ) ∖ ( 𝐶 ∖ 𝐴 ) )

Proof

Step Hyp Ref Expression
1 elun ⊢ ( 𝑥 ∈ ( 𝐴 ∪ 𝐵 ) ↔ ( 𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵 ) )
2 pm4.53 ⊢ ( ¬ ( 𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐴 ) ↔ ( ¬ 𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐴 ) )
3 eldif ⊢ ( 𝑥 ∈ ( 𝐶 ∖ 𝐴 ) ↔ ( 𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐴 ) )
4 2 3 xchnxbir ⊢ ( ¬ 𝑥 ∈ ( 𝐶 ∖ 𝐴 ) ↔ ( ¬ 𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐴 ) )
5 1 4 anbi12i ⊢ ( ( 𝑥 ∈ ( 𝐴 ∪ 𝐵 ) ∧ ¬ 𝑥 ∈ ( 𝐶 ∖ 𝐴 ) ) ↔ ( ( 𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵 ) ∧ ( ¬ 𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐴 ) ) )
6 eldif ⊢ ( 𝑥 ∈ ( ( 𝐴 ∪ 𝐵 ) ∖ ( 𝐶 ∖ 𝐴 ) ) ↔ ( 𝑥 ∈ ( 𝐴 ∪ 𝐵 ) ∧ ¬ 𝑥 ∈ ( 𝐶 ∖ 𝐴 ) ) )
7 elun ⊢ ( 𝑥 ∈ ( 𝐴 ∪ ( 𝐵 ∖ 𝐶 ) ) ↔ ( 𝑥 ∈ 𝐴 ∨ 𝑥 ∈ ( 𝐵 ∖ 𝐶 ) ) )
8 eldif ⊢ ( 𝑥 ∈ ( 𝐵 ∖ 𝐶 ) ↔ ( 𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶 ) )
9 8 orbi2i ⊢ ( ( 𝑥 ∈ 𝐴 ∨ 𝑥 ∈ ( 𝐵 ∖ 𝐶 ) ) ↔ ( 𝑥 ∈ 𝐴 ∨ ( 𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶 ) ) )
10 ordi ⊢ ( ( 𝑥 ∈ 𝐴 ∨ ( 𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶 ) ) ↔ ( ( 𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 ∨ ¬ 𝑥 ∈ 𝐶 ) ) )
11 orcom ⊢ ( ( 𝑥 ∈ 𝐴 ∨ ¬ 𝑥 ∈ 𝐶 ) ↔ ( ¬ 𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐴 ) )
12 11 anbi2i ⊢ ( ( ( 𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 ∨ ¬ 𝑥 ∈ 𝐶 ) ) ↔ ( ( 𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵 ) ∧ ( ¬ 𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐴 ) ) )
13 10 12 bitri ⊢ ( ( 𝑥 ∈ 𝐴 ∨ ( 𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶 ) ) ↔ ( ( 𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵 ) ∧ ( ¬ 𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐴 ) ) )
14 7 9 13 3bitri ⊢ ( 𝑥 ∈ ( 𝐴 ∪ ( 𝐵 ∖ 𝐶 ) ) ↔ ( ( 𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵 ) ∧ ( ¬ 𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐴 ) ) )
15 5 6 14 3bitr4ri ⊢ ( 𝑥 ∈ ( 𝐴 ∪ ( 𝐵 ∖ 𝐶 ) ) ↔ 𝑥 ∈ ( ( 𝐴 ∪ 𝐵 ) ∖ ( 𝐶 ∖ 𝐴 ) ) )
16 15 eqriv ⊢ ( 𝐴 ∪ ( 𝐵 ∖ 𝐶 ) ) = ( ( 𝐴 ∪ 𝐵 ) ∖ ( 𝐶 ∖ 𝐴 ) )