![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > psssstrd | Structured version Visualization version GIF version |
Description: Transitivity involving subclass and proper subclass inclusion. Deduction form of psssstr 4106. (Contributed by David Moews, 1-May-2017.) |
Ref | Expression |
---|---|
psssstrd.1 | ⊢ (𝜑 → 𝐴 ⊊ 𝐵) |
psssstrd.2 | ⊢ (𝜑 → 𝐵 ⊆ 𝐶) |
Ref | Expression |
---|---|
psssstrd | ⊢ (𝜑 → 𝐴 ⊊ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | psssstrd.1 | . 2 ⊢ (𝜑 → 𝐴 ⊊ 𝐵) | |
2 | psssstrd.2 | . 2 ⊢ (𝜑 → 𝐵 ⊆ 𝐶) | |
3 | psssstr 4106 | . 2 ⊢ ((𝐴 ⊊ 𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐴 ⊊ 𝐶) | |
4 | 1, 2, 3 | syl2anc 582 | 1 ⊢ (𝜑 → 𝐴 ⊊ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ⊆ wss 3949 ⊊ wpss 3950 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2706 df-cleq 2720 df-clel 2806 df-ne 2938 df-v 3475 df-in 3956 df-ss 3966 df-pss 3968 |
This theorem is referenced by: ackbij1lem15 10265 lsatssn0 38506 lsatexch 38547 lsatcvatlem 38553 lkrpssN 38667 |
Copyright terms: Public domain | W3C validator |