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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 4473. (Contributed by David Moews, 1-May-2017.) |
Ref | Expression |
---|---|
difsnb | ⊢ (¬ 𝐴 ∈ 𝐵 ↔ (𝐵 ∖ {𝐴}) = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | difsn 4473 | . 2 ⊢ (¬ 𝐴 ∈ 𝐵 → (𝐵 ∖ {𝐴}) = 𝐵) | |
2 | neldifsnd 4468 | . . . . 5 ⊢ (𝐴 ∈ 𝐵 → ¬ 𝐴 ∈ (𝐵 ∖ {𝐴})) | |
3 | nelne1 3028 | . . . . 5 ⊢ ((𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ (𝐵 ∖ {𝐴})) → 𝐵 ≠ (𝐵 ∖ {𝐴})) | |
4 | 2, 3 | mpdan 705 | . . . 4 ⊢ (𝐴 ∈ 𝐵 → 𝐵 ≠ (𝐵 ∖ {𝐴})) |
5 | 4 | necomd 2987 | . . 3 ⊢ (𝐴 ∈ 𝐵 → (𝐵 ∖ {𝐴}) ≠ 𝐵) |
6 | 5 | necon2bi 2962 | . 2 ⊢ ((𝐵 ∖ {𝐴}) = 𝐵 → ¬ 𝐴 ∈ 𝐵) |
7 | 1, 6 | impbii 199 | 1 ⊢ (¬ 𝐴 ∈ 𝐵 ↔ (𝐵 ∖ {𝐴}) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 196 = wceq 1632 ∈ wcel 2139 ≠ wne 2932 ∖ cdif 3712 {csn 4321 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-v 3342 df-dif 3718 df-sn 4322 |
This theorem is referenced by: difsnpss 4483 incexclem 14787 mrieqv2d 16521 mreexmrid 16525 mreexexlem2d 16527 mreexexlem4d 16529 acsfiindd 17398 |
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