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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 4797. (Contributed by David Moews, 1-May-2017.) | 
| Ref | Expression | 
|---|---|
| difsnb | ⊢ (¬ 𝐴 ∈ 𝐵 ↔ (𝐵 ∖ {𝐴}) = 𝐵) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | difsn 4797 | . 2 ⊢ (¬ 𝐴 ∈ 𝐵 → (𝐵 ∖ {𝐴}) = 𝐵) | |
| 2 | neldifsnd 4792 | . . . . 5 ⊢ (𝐴 ∈ 𝐵 → ¬ 𝐴 ∈ (𝐵 ∖ {𝐴})) | |
| 3 | nelne1 3038 | . . . . 5 ⊢ ((𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ (𝐵 ∖ {𝐴})) → 𝐵 ≠ (𝐵 ∖ {𝐴})) | |
| 4 | 2, 3 | mpdan 687 | . . . 4 ⊢ (𝐴 ∈ 𝐵 → 𝐵 ≠ (𝐵 ∖ {𝐴})) | 
| 5 | 4 | necomd 2995 | . . 3 ⊢ (𝐴 ∈ 𝐵 → (𝐵 ∖ {𝐴}) ≠ 𝐵) | 
| 6 | 5 | necon2bi 2970 | . 2 ⊢ ((𝐵 ∖ {𝐴}) = 𝐵 → ¬ 𝐴 ∈ 𝐵) | 
| 7 | 1, 6 | impbii 209 | 1 ⊢ (¬ 𝐴 ∈ 𝐵 ↔ (𝐵 ∖ {𝐴}) = 𝐵) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 ↔ wb 206 = wceq 1539 ∈ wcel 2107 ≠ wne 2939 ∖ cdif 3947 {csn 4625 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1542 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-v 3481 df-dif 3953 df-sn 4626 | 
| This theorem is referenced by: difsnpss 4806 incexclem 15873 mrieqv2d 17683 mreexmrid 17687 mreexexlem2d 17689 mreexexlem4d 17691 acsfiindd 18599 | 
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