Description: Obsolete version of sstr2 as of 19-May-2025. (Contributed by NM, 24-Jun-1993) (Proof shortened by Andrew Salmon, 14-Jun-2011) (Proof modification is discouraged.) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | sstr2OLD | ⊢ ( 𝐴 ⊆ 𝐵 → ( 𝐵 ⊆ 𝐶 → 𝐴 ⊆ 𝐶 ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ssel | ⊢ ( 𝐴 ⊆ 𝐵 → ( 𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵 ) ) | |
| 2 | 1 | imim1d | ⊢ ( 𝐴 ⊆ 𝐵 → ( ( 𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐶 ) → ( 𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶 ) ) ) |
| 3 | 2 | alimdv | ⊢ ( 𝐴 ⊆ 𝐵 → ( ∀ 𝑥 ( 𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐶 ) → ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶 ) ) ) |
| 4 | df-ss | ⊢ ( 𝐵 ⊆ 𝐶 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐶 ) ) | |
| 5 | df-ss | ⊢ ( 𝐴 ⊆ 𝐶 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶 ) ) | |
| 6 | 3 4 5 | 3imtr4g | ⊢ ( 𝐴 ⊆ 𝐵 → ( 𝐵 ⊆ 𝐶 → 𝐴 ⊆ 𝐶 ) ) |