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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 4001 | . . 3 ⊢ (𝐴 = 𝐵 → 𝐴 ⊆ 𝐵) | |
2 | ssdif0 4324 | . . 3 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∖ 𝐵) = ∅) | |
3 | 1, 2 | sylib 217 | . 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∖ 𝐵) = ∅) |
4 | 3 | necon3i 2973 | 1 ⊢ ((𝐴 ∖ 𝐵) ≠ ∅ → 𝐴 ≠ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = 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: disjdsct 31663 bj-2upln1upl 35541 |
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