Metamath Proof Explorer


Theorem ifr0

Description: A class that is founded by the identity relation is null. (Contributed by Andrew Salmon, 25-Jul-2011)

Ref Expression
Assertion ifr0 ( I Fr 𝐴𝐴 = ∅ )

Proof

Step Hyp Ref Expression
1 equid 𝑥 = 𝑥
2 vex 𝑥 ∈ V
3 2 ideq ( 𝑥 I 𝑥𝑥 = 𝑥 )
4 1 3 mpbir 𝑥 I 𝑥
5 frirr ( ( I Fr 𝐴𝑥𝐴 ) → ¬ 𝑥 I 𝑥 )
6 5 ex ( I Fr 𝐴 → ( 𝑥𝐴 → ¬ 𝑥 I 𝑥 ) )
7 4 6 mt2i ( I Fr 𝐴 → ¬ 𝑥𝐴 )
8 7 eq0rdv ( I Fr 𝐴𝐴 = ∅ )
9 fr0 I Fr ∅
10 freq2 ( 𝐴 = ∅ → ( I Fr 𝐴 ↔ I Fr ∅ ) )
11 9 10 mpbiri ( 𝐴 = ∅ → I Fr 𝐴 )
12 8 11 impbii ( I Fr 𝐴𝐴 = ∅ )