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Mirrors > Home > ILE Home > Th. List > eldifsn | GIF version |
Description: Membership in a set with an element removed. (Contributed by NM, 10-Oct-2007.) |
Ref | Expression |
---|---|
eldifsn | ⊢ (𝐴 ∈ (𝐵 ∖ {𝐶}) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ≠ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldif 3080 | . 2 ⊢ (𝐴 ∈ (𝐵 ∖ {𝐶}) ↔ (𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ {𝐶})) | |
2 | elsng 3542 | . . . 4 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ {𝐶} ↔ 𝐴 = 𝐶)) | |
3 | 2 | necon3bbid 2348 | . . 3 ⊢ (𝐴 ∈ 𝐵 → (¬ 𝐴 ∈ {𝐶} ↔ 𝐴 ≠ 𝐶)) |
4 | 3 | pm5.32i 449 | . 2 ⊢ ((𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ {𝐶}) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ≠ 𝐶)) |
5 | 1, 4 | bitri 183 | 1 ⊢ (𝐴 ∈ (𝐵 ∖ {𝐶}) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ≠ 𝐶)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∧ wa 103 ↔ wb 104 ∈ wcel 1480 ≠ wne 2308 ∖ cdif 3068 {csn 3527 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-v 2688 df-dif 3073 df-sn 3533 |
This theorem is referenced by: eldifsni 3652 rexdifsn 3655 difsn 3657 fnniniseg2 5543 rexsupp 5544 mpodifsnif 5864 suppssfv 5978 suppssov1 5979 dif1o 6335 fidifsnen 6764 en2eleq 7051 en2other2 7052 elni 7116 divvalap 8434 elnnne0 8991 divfnzn 9413 modfzo0difsn 10168 modsumfzodifsn 10169 hashdifpr 10566 eff2 11386 tanvalap 11415 fzo0dvdseq 11555 oddprmgt2 11814 setsslnid 12010 |
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