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Mirrors > Home > MPE Home > Th. List > pssdifn0 | Structured version Visualization version GIF version |
Description: A proper subclass has a nonempty difference. (Contributed by NM, 3-May-1994.) |
Ref | Expression |
---|---|
pssdifn0 | ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ 𝐵) → (𝐵 ∖ 𝐴) ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssdif0 4324 | . . . 4 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐵 ∖ 𝐴) = ∅) | |
2 | eqss 3960 | . . . . 5 ⊢ (𝐴 = 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴)) | |
3 | 2 | simplbi2 502 | . . . 4 ⊢ (𝐴 ⊆ 𝐵 → (𝐵 ⊆ 𝐴 → 𝐴 = 𝐵)) |
4 | 1, 3 | biimtrrid 242 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → ((𝐵 ∖ 𝐴) = ∅ → 𝐴 = 𝐵)) |
5 | 4 | necon3d 2961 | . 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ≠ 𝐵 → (𝐵 ∖ 𝐴) ≠ ∅)) |
6 | 5 | imp 408 | 1 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ 𝐵) → (𝐵 ∖ 𝐴) ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ≠ wne 2940 ∖ cdif 3908 ⊆ wss 3911 ∅c0 4283 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2941 df-v 3446 df-dif 3914 df-in 3918 df-ss 3928 df-nul 4284 |
This theorem is referenced by: pssdif 4327 tz7.7 6344 domdifsn 9001 inf3lem3 9571 isf32lem6 10299 fclscf 23392 flimfnfcls 23395 lebnumlem1 24340 lebnumlem2 24341 lebnumlem3 24342 ig1peu 25552 ig1pdvds 25557 qsidomlem2 32274 divrngidl 36533 |
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