| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > sylan9ssr | Structured version Visualization version GIF version | ||
| Description: A subclass transitivity deduction. (Contributed by NM, 27-Sep-2004.) |
| Ref | Expression |
|---|---|
| sylan9ssr.1 | ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
| sylan9ssr.2 | ⊢ (𝜓 → 𝐵 ⊆ 𝐶) |
| Ref | Expression |
|---|---|
| sylan9ssr | ⊢ ((𝜓 ∧ 𝜑) → 𝐴 ⊆ 𝐶) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | sylan9ssr.1 | . . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | |
| 2 | sylan9ssr.2 | . . 3 ⊢ (𝜓 → 𝐵 ⊆ 𝐶) | |
| 3 | 1, 2 | sylan9ss 3997 | . 2 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 ⊆ 𝐶) |
| 4 | 3 | ancoms 458 | 1 ⊢ ((𝜓 ∧ 𝜑) → 𝐴 ⊆ 𝐶) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ⊆ wss 3951 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ss 3968 |
| This theorem is referenced by: intssuni2 4973 marypha1 9474 cardinfima 10137 cfflb 10299 ssfin4 10350 acsfn 17702 mrelatlub 18607 efgval 19735 islbs3 21157 kgentopon 23546 txlly 23644 sigaclci 34133 bnj1014 34975 topjoin 36366 filnetlem3 36381 poimirlem16 37643 mblfinlem3 37666 sspwimpALT 44945 sspwimpALT2 44948 setrecsres 49221 |
| Copyright terms: Public domain | W3C validator |