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Mirrors > Home > ILE Home > Th. List > nelpri | GIF version |
Description: If an element doesn't match the items in an unordered pair, it is not in the unordered pair. (Contributed by David A. Wheeler, 10-May-2015.) |
Ref | Expression |
---|---|
nelpri.1 | ⊢ 𝐴 ≠ 𝐵 |
nelpri.2 | ⊢ 𝐴 ≠ 𝐶 |
Ref | Expression |
---|---|
nelpri | ⊢ ¬ 𝐴 ∈ {𝐵, 𝐶} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nelpri.1 | . 2 ⊢ 𝐴 ≠ 𝐵 | |
2 | nelpri.2 | . 2 ⊢ 𝐴 ≠ 𝐶 | |
3 | neanior 2393 | . . 3 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ↔ ¬ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶)) | |
4 | elpri 3545 | . . . 4 ⊢ (𝐴 ∈ {𝐵, 𝐶} → (𝐴 = 𝐵 ∨ 𝐴 = 𝐶)) | |
5 | 4 | con3i 621 | . . 3 ⊢ (¬ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶) → ¬ 𝐴 ∈ {𝐵, 𝐶}) |
6 | 3, 5 | sylbi 120 | . 2 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) → ¬ 𝐴 ∈ {𝐵, 𝐶}) |
7 | 1, 2, 6 | mp2an 422 | 1 ⊢ ¬ 𝐴 ∈ {𝐵, 𝐶} |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∧ wa 103 ∨ wo 697 = wceq 1331 ∈ wcel 1480 ≠ wne 2306 {cpr 3523 |
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 2119 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-v 2683 df-un 3070 df-sn 3528 df-pr 3529 |
This theorem is referenced by: prneli 3547 |
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