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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 4358 | . . . 4 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐵 ∖ 𝐴) = ∅) | |
2 | eqss 3992 | . . . . 5 ⊢ (𝐴 = 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴)) | |
3 | 2 | simplbi2 500 | . . . 4 ⊢ (𝐴 ⊆ 𝐵 → (𝐵 ⊆ 𝐴 → 𝐴 = 𝐵)) |
4 | 1, 3 | biimtrrid 242 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → ((𝐵 ∖ 𝐴) = ∅ → 𝐴 = 𝐵)) |
5 | 4 | necon3d 2955 | . 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ≠ 𝐵 → (𝐵 ∖ 𝐴) ≠ ∅)) |
6 | 5 | imp 406 | 1 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ 𝐵) → (𝐵 ∖ 𝐴) ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ≠ wne 2934 ∖ cdif 3940 ⊆ wss 3943 ∅c0 4317 |
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 2697 |
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 2704 df-cleq 2718 df-clel 2804 df-ne 2935 df-v 3470 df-dif 3946 df-in 3950 df-ss 3960 df-nul 4318 |
This theorem is referenced by: pssdif 4361 tz7.7 6384 domdifsn 9056 inf3lem3 9627 isf32lem6 10355 fclscf 23884 flimfnfcls 23887 lebnumlem1 24842 lebnumlem2 24843 lebnumlem3 24844 ig1peu 26064 ig1pdvds 26069 qsidomlem2 33078 qsdrng 33117 divrngidl 37409 |
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