![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > pssdif | Structured version Visualization version GIF version |
Description: A proper subclass has a nonempty difference. (Contributed by Mario Carneiro, 27-Apr-2016.) |
Ref | Expression |
---|---|
pssdif | ⊢ (𝐴 ⊊ 𝐵 → (𝐵 ∖ 𝐴) ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-pss 3959 | . 2 ⊢ (𝐴 ⊊ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ 𝐵)) | |
2 | pssdifn0 4357 | . 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ 𝐵) → (𝐵 ∖ 𝐴) ≠ ∅) | |
3 | 1, 2 | sylbi 216 | 1 ⊢ (𝐴 ⊊ 𝐵 → (𝐵 ∖ 𝐴) ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ≠ wne 2932 ∖ cdif 3937 ⊆ wss 3940 ⊊ wpss 3941 ∅c0 4314 |
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 2695 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-ne 2933 df-v 3468 df-dif 3943 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 |
This theorem is referenced by: pssnel 4462 pgpfac1lem5 19991 fundmpss 35233 dfon2lem6 35255 |
Copyright terms: Public domain | W3C validator |