Metamath Proof Explorer


Theorem srcmpltd

Description: If a statement is true for every element of a class and for every element of its complement relative to a second class, then it is true for every element in the second class. (Contributed by BTernaryTau, 27-Sep-2023)

Ref Expression
Hypotheses srcmpltd.1 ( 𝜑 → ( 𝐶𝐴𝜓 ) )
srcmpltd.2 ( 𝜑 → ( 𝐶 ∈ ( 𝐵𝐴 ) → 𝜓 ) )
Assertion srcmpltd ( 𝜑 → ( 𝐶𝐵𝜓 ) )

Proof

Step Hyp Ref Expression
1 srcmpltd.1 ( 𝜑 → ( 𝐶𝐴𝜓 ) )
2 srcmpltd.2 ( 𝜑 → ( 𝐶 ∈ ( 𝐵𝐴 ) → 𝜓 ) )
3 elun2 ( 𝐶𝐵𝐶 ∈ ( 𝐴𝐵 ) )
4 undif2 ( 𝐴 ∪ ( 𝐵𝐴 ) ) = ( 𝐴𝐵 )
5 3 4 eleqtrrdi ( 𝐶𝐵𝐶 ∈ ( 𝐴 ∪ ( 𝐵𝐴 ) ) )
6 elunant ( ( 𝐶 ∈ ( 𝐴 ∪ ( 𝐵𝐴 ) ) → 𝜓 ) ↔ ( ( 𝐶𝐴𝜓 ) ∧ ( 𝐶 ∈ ( 𝐵𝐴 ) → 𝜓 ) ) )
7 1 2 6 sylanbrc ( 𝜑 → ( 𝐶 ∈ ( 𝐴 ∪ ( 𝐵𝐴 ) ) → 𝜓 ) )
8 5 7 syl5 ( 𝜑 → ( 𝐶𝐵𝜓 ) )