Description: Any nonempty transitive class includes its intersection. Exercise 3 in TakeutiZaring p. 44 (which mistakenly does not include the nonemptiness hypothesis). (Contributed by Scott Fenton, 3-Mar-2011) (Proof shortened by Andrew Salmon, 14-Nov-2011)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | trintss | ⊢ ( ( Tr 𝐴 ∧ 𝐴 ≠ ∅ ) → ∩ 𝐴 ⊆ 𝐴 ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | n0 | ⊢ ( 𝐴 ≠ ∅ ↔ ∃ 𝑥 𝑥 ∈ 𝐴 ) | |
| 2 | intss1 | ⊢ ( 𝑥 ∈ 𝐴 → ∩ 𝐴 ⊆ 𝑥 ) | |
| 3 | trss | ⊢ ( Tr 𝐴 → ( 𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐴 ) ) | |
| 4 | 3 | com12 | ⊢ ( 𝑥 ∈ 𝐴 → ( Tr 𝐴 → 𝑥 ⊆ 𝐴 ) ) |
| 5 | sstr2 | ⊢ ( ∩ 𝐴 ⊆ 𝑥 → ( 𝑥 ⊆ 𝐴 → ∩ 𝐴 ⊆ 𝐴 ) ) | |
| 6 | 2 4 5 | sylsyld | ⊢ ( 𝑥 ∈ 𝐴 → ( Tr 𝐴 → ∩ 𝐴 ⊆ 𝐴 ) ) |
| 7 | 6 | exlimiv | ⊢ ( ∃ 𝑥 𝑥 ∈ 𝐴 → ( Tr 𝐴 → ∩ 𝐴 ⊆ 𝐴 ) ) |
| 8 | 1 7 | sylbi | ⊢ ( 𝐴 ≠ ∅ → ( Tr 𝐴 → ∩ 𝐴 ⊆ 𝐴 ) ) |
| 9 | 8 | impcom | ⊢ ( ( Tr 𝐴 ∧ 𝐴 ≠ ∅ ) → ∩ 𝐴 ⊆ 𝐴 ) |