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Mirrors > Home > ILE Home > Th. List > difsnb | GIF version |
Description: (𝐵 ∖ {𝐴}) equals 𝐵 if and only if 𝐴 is not a member of 𝐵. Generalization of difsn 3726. (Contributed by David Moews, 1-May-2017.) |
Ref | Expression |
---|---|
difsnb | ⊢ (¬ 𝐴 ∈ 𝐵 ↔ (𝐵 ∖ {𝐴}) = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | difsn 3726 | . 2 ⊢ (¬ 𝐴 ∈ 𝐵 → (𝐵 ∖ {𝐴}) = 𝐵) | |
2 | neldifsnd 3720 | . . . . 5 ⊢ (𝐴 ∈ 𝐵 → ¬ 𝐴 ∈ (𝐵 ∖ {𝐴})) | |
3 | nelne1 2435 | . . . . 5 ⊢ ((𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ (𝐵 ∖ {𝐴})) → 𝐵 ≠ (𝐵 ∖ {𝐴})) | |
4 | 2, 3 | mpdan 421 | . . . 4 ⊢ (𝐴 ∈ 𝐵 → 𝐵 ≠ (𝐵 ∖ {𝐴})) |
5 | 4 | necomd 2431 | . . 3 ⊢ (𝐴 ∈ 𝐵 → (𝐵 ∖ {𝐴}) ≠ 𝐵) |
6 | 5 | necon2bi 2400 | . 2 ⊢ ((𝐵 ∖ {𝐴}) = 𝐵 → ¬ 𝐴 ∈ 𝐵) |
7 | 1, 6 | impbii 126 | 1 ⊢ (¬ 𝐴 ∈ 𝐵 ↔ (𝐵 ∖ {𝐴}) = 𝐵) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ↔ wb 105 = wceq 1353 ∈ wcel 2146 ≠ wne 2345 ∖ cdif 3124 {csn 3589 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1445 ax-7 1446 ax-gen 1447 ax-ie1 1491 ax-ie2 1492 ax-8 1502 ax-10 1503 ax-11 1504 ax-i12 1505 ax-bndl 1507 ax-4 1508 ax-17 1524 ax-i9 1528 ax-ial 1532 ax-i5r 1533 ax-ext 2157 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1459 df-sb 1761 df-clab 2162 df-cleq 2168 df-clel 2171 df-nfc 2306 df-ne 2346 df-v 2737 df-dif 3129 df-sn 3595 |
This theorem is referenced by: (None) |
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