| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > sseqtri | Structured version Visualization version GIF version | ||
| Description: Substitution of equality into a subclass relationship. (Contributed by NM, 28-Jul-1995.) |
| Ref | Expression |
|---|---|
| sseqtr.1 | ⊢ 𝐴 ⊆ 𝐵 |
| sseqtr.2 | ⊢ 𝐵 = 𝐶 |
| Ref | Expression |
|---|---|
| sseqtri | ⊢ 𝐴 ⊆ 𝐶 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | sseqtr.1 | . 2 ⊢ 𝐴 ⊆ 𝐵 | |
| 2 | sseqtr.2 | . . 3 ⊢ 𝐵 = 𝐶 | |
| 3 | 2 | sseq2i 4013 | . 2 ⊢ (𝐴 ⊆ 𝐵 ↔ 𝐴 ⊆ 𝐶) |
| 4 | 1, 3 | mpbi 230 | 1 ⊢ 𝐴 ⊆ 𝐶 |
| Copyright terms: Public domain | W3C validator |