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Mirrors > Home > MPE Home > Th. List > difn0 | Structured version Visualization version GIF version |
Description: If the difference of two sets is not empty, then the sets are not equal. (Contributed by Thierry Arnoux, 28-Feb-2017.) |
Ref | Expression |
---|---|
difn0 | ⊢ ((𝐴 ∖ 𝐵) ≠ ∅ → 𝐴 ≠ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqimss 3943 | . . 3 ⊢ (𝐴 = 𝐵 → 𝐴 ⊆ 𝐵) | |
2 | ssdif0 4264 | . . 3 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∖ 𝐵) = ∅) | |
3 | 1, 2 | sylib 221 | . 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∖ 𝐵) = ∅) |
4 | 3 | necon3i 2964 | 1 ⊢ ((𝐴 ∖ 𝐵) ≠ ∅ → 𝐴 ≠ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ≠ wne 2932 ∖ cdif 3850 ⊆ wss 3853 ∅c0 4223 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-ne 2933 df-v 3400 df-dif 3856 df-in 3860 df-ss 3870 df-nul 4224 |
This theorem is referenced by: disjdsct 30709 bj-2upln1upl 34900 |
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