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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 3852 | . . 3 ⊢ (𝐴 = 𝐵 → 𝐴 ⊆ 𝐵) | |
2 | ssdif0 4141 | . . 3 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∖ 𝐵) = ∅) | |
3 | 1, 2 | sylib 210 | . 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∖ 𝐵) = ∅) |
4 | 3 | necon3i 3002 | 1 ⊢ ((𝐴 ∖ 𝐵) ≠ ∅ → 𝐴 ≠ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1653 ≠ wne 2970 ∖ cdif 3765 ⊆ wss 3768 ∅c0 4114 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2776 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-clab 2785 df-cleq 2791 df-clel 2794 df-nfc 2929 df-ne 2971 df-v 3386 df-dif 3771 df-in 3775 df-ss 3782 df-nul 4115 |
This theorem is referenced by: disjdsct 29991 bj-2upln1upl 33497 |
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