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Mirrors > Home > MPE Home > Th. List > difsnb | Structured version Visualization version GIF version |
Description: (𝐵 ∖ {𝐴}) equals 𝐵 if and only if 𝐴 is not a member of 𝐵. Generalization of difsn 4733. (Contributed by David Moews, 1-May-2017.) |
Ref | Expression |
---|---|
difsnb | ⊢ (¬ 𝐴 ∈ 𝐵 ↔ (𝐵 ∖ {𝐴}) = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | difsn 4733 | . 2 ⊢ (¬ 𝐴 ∈ 𝐵 → (𝐵 ∖ {𝐴}) = 𝐵) | |
2 | neldifsnd 4728 | . . . . 5 ⊢ (𝐴 ∈ 𝐵 → ¬ 𝐴 ∈ (𝐵 ∖ {𝐴})) | |
3 | nelne1 3115 | . . . . 5 ⊢ ((𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ (𝐵 ∖ {𝐴})) → 𝐵 ≠ (𝐵 ∖ {𝐴})) | |
4 | 2, 3 | mpdan 685 | . . . 4 ⊢ (𝐴 ∈ 𝐵 → 𝐵 ≠ (𝐵 ∖ {𝐴})) |
5 | 4 | necomd 3073 | . . 3 ⊢ (𝐴 ∈ 𝐵 → (𝐵 ∖ {𝐴}) ≠ 𝐵) |
6 | 5 | necon2bi 3048 | . 2 ⊢ ((𝐵 ∖ {𝐴}) = 𝐵 → ¬ 𝐴 ∈ 𝐵) |
7 | 1, 6 | impbii 211 | 1 ⊢ (¬ 𝐴 ∈ 𝐵 ↔ (𝐵 ∖ {𝐴}) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 208 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 ∖ cdif 3935 {csn 4569 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-v 3498 df-dif 3941 df-sn 4570 |
This theorem is referenced by: difsnpss 4742 incexclem 15193 mrieqv2d 16912 mreexmrid 16916 mreexexlem2d 16918 mreexexlem4d 16920 acsfiindd 17789 |
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