Metamath Proof Explorer


Theorem onnminsb

Description: An ordinal number smaller than the minimum of a set of ordinal numbers does not have the property determining that set. ps is the wff resulting from the substitution of A for x in wff ph . (Contributed by NM, 9-Nov-2003)

Ref Expression
Hypothesis onnminsb.1 ( 𝑥 = 𝐴 → ( 𝜑𝜓 ) )
Assertion onnminsb ( 𝐴 ∈ On → ( 𝐴 { 𝑥 ∈ On ∣ 𝜑 } → ¬ 𝜓 ) )

Proof

Step Hyp Ref Expression
1 onnminsb.1 ( 𝑥 = 𝐴 → ( 𝜑𝜓 ) )
2 1 elrab ( 𝐴 ∈ { 𝑥 ∈ On ∣ 𝜑 } ↔ ( 𝐴 ∈ On ∧ 𝜓 ) )
3 ssrab2 { 𝑥 ∈ On ∣ 𝜑 } ⊆ On
4 onnmin ( ( { 𝑥 ∈ On ∣ 𝜑 } ⊆ On ∧ 𝐴 ∈ { 𝑥 ∈ On ∣ 𝜑 } ) → ¬ 𝐴 { 𝑥 ∈ On ∣ 𝜑 } )
5 3 4 mpan ( 𝐴 ∈ { 𝑥 ∈ On ∣ 𝜑 } → ¬ 𝐴 { 𝑥 ∈ On ∣ 𝜑 } )
6 2 5 sylbir ( ( 𝐴 ∈ On ∧ 𝜓 ) → ¬ 𝐴 { 𝑥 ∈ On ∣ 𝜑 } )
7 6 ex ( 𝐴 ∈ On → ( 𝜓 → ¬ 𝐴 { 𝑥 ∈ On ∣ 𝜑 } ) )
8 7 con2d ( 𝐴 ∈ On → ( 𝐴 { 𝑥 ∈ On ∣ 𝜑 } → ¬ 𝜓 ) )