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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 3900 | . 2 ⊢ (𝐴 ⊊ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ 𝐵)) | |
2 | pssdifn0 4279 | . 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ 𝐵) → (𝐵 ∖ 𝐴) ≠ ∅) | |
3 | 1, 2 | sylbi 220 | 1 ⊢ (𝐴 ⊊ 𝐵 → (𝐵 ∖ 𝐴) ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ≠ wne 2987 ∖ cdif 3878 ⊆ wss 3881 ⊊ wpss 3882 ∅c0 4243 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-ne 2988 df-v 3443 df-dif 3884 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 |
This theorem is referenced by: pssnel 4378 pgpfac1lem5 19194 fundmpss 33122 dfon2lem6 33146 |
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