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Mirrors > Home > MPE Home > Th. List > difsn | Structured version Visualization version GIF version |
Description: An element not in a set can be removed without affecting the set. (Contributed by NM, 16-Mar-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
Ref | Expression |
---|---|
difsn | ⊢ (¬ 𝐴 ∈ 𝐵 → (𝐵 ∖ {𝐴}) = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldifsn 4711 | . . 3 ⊢ (𝑥 ∈ (𝐵 ∖ {𝐴}) ↔ (𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 𝐴)) | |
2 | simpl 483 | . . . 4 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 𝐴) → 𝑥 ∈ 𝐵) | |
3 | nelelne 3114 | . . . . 5 ⊢ (¬ 𝐴 ∈ 𝐵 → (𝑥 ∈ 𝐵 → 𝑥 ≠ 𝐴)) | |
4 | 3 | ancld 551 | . . . 4 ⊢ (¬ 𝐴 ∈ 𝐵 → (𝑥 ∈ 𝐵 → (𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 𝐴))) |
5 | 2, 4 | impbid2 227 | . . 3 ⊢ (¬ 𝐴 ∈ 𝐵 → ((𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 𝐴) ↔ 𝑥 ∈ 𝐵)) |
6 | 1, 5 | syl5bb 284 | . 2 ⊢ (¬ 𝐴 ∈ 𝐵 → (𝑥 ∈ (𝐵 ∖ {𝐴}) ↔ 𝑥 ∈ 𝐵)) |
7 | 6 | eqrdv 2816 | 1 ⊢ (¬ 𝐴 ∈ 𝐵 → (𝐵 ∖ {𝐴}) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 ∖ cdif 3930 {csn 4557 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-v 3494 df-dif 3936 df-sn 4558 |
This theorem is referenced by: difsnb 4731 difsnexi 7472 domdifsn 8588 domunsncan 8605 frfi 8751 infdifsn 9108 dfn2 11898 hashgt23el 13773 clslp 21684 xrge00 30600 lindsadd 34766 lindsenlbs 34768 poimirlem2 34775 poimirlem4 34777 poimirlem6 34779 poimirlem7 34780 poimirlem8 34781 poimirlem19 34792 poimirlem23 34796 supxrmnf2 41583 infxrpnf2 41615 dvmptfprodlem 42105 hoiprodp1 42747 |
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