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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 3956 | . 2 ⊢ (𝐴 ⊊ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ 𝐵)) | |
2 | pssdifn0 4327 | . 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ 𝐵) → (𝐵 ∖ 𝐴) ≠ ∅) | |
3 | 1, 2 | sylbi 219 | 1 ⊢ (𝐴 ⊊ 𝐵 → (𝐵 ∖ 𝐴) ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ≠ wne 3018 ∖ cdif 3935 ⊆ wss 3938 ⊊ wpss 3939 ∅c0 4293 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-v 3498 df-dif 3941 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 |
This theorem is referenced by: pssnel 4422 pgpfac1lem5 19203 fundmpss 33011 dfon2lem6 33035 |
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