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 → ( 𝜑 ↔ 𝜓 ) ) | |
| Assertion | onnminsb | ⊢ ( 𝐴 ∈ On → ( 𝐴 ∈ ∩ { 𝑥 ∈ On ∣ 𝜑 } → ¬ 𝜓 ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | onnminsb.1 | ⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) ) | |
| 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 ) ) |