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| 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 3971 | . 2 ⊢ (𝐴 ⊊ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ 𝐵)) | |
| 2 | pssdifn0 4368 | . 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ 𝐵) → (𝐵 ∖ 𝐴) ≠ ∅) | |
| 3 | 1, 2 | sylbi 217 | 1 ⊢ (𝐴 ⊊ 𝐵 → (𝐵 ∖ 𝐴) ≠ ∅) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ≠ wne 2940 ∖ cdif 3948 ⊆ wss 3951 ⊊ wpss 3952 ∅c0 4333 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-v 3482 df-dif 3954 df-ss 3968 df-pss 3971 df-nul 4334 |
| This theorem is referenced by: pssnel 4471 pgpfac1lem5 20099 fundmpss 35767 dfon2lem6 35789 |
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