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Mirrors > Home > ILE Home > Th. List > prneli | GIF version |
Description: If an element doesn't match the items in an unordered pair, it is not in the unordered pair, using ∉. (Contributed by David A. Wheeler, 10-May-2015.) |
Ref | Expression |
---|---|
prneli.1 | ⊢ 𝐴 ≠ 𝐵 |
prneli.2 | ⊢ 𝐴 ≠ 𝐶 |
Ref | Expression |
---|---|
prneli | ⊢ 𝐴 ∉ {𝐵, 𝐶} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prneli.1 | . . 3 ⊢ 𝐴 ≠ 𝐵 | |
2 | prneli.2 | . . 3 ⊢ 𝐴 ≠ 𝐶 | |
3 | 1, 2 | nelpri 3551 | . 2 ⊢ ¬ 𝐴 ∈ {𝐵, 𝐶} |
4 | 3 | nelir 2406 | 1 ⊢ 𝐴 ∉ {𝐵, 𝐶} |
Colors of variables: wff set class |
Syntax hints: ≠ wne 2308 ∉ wnel 2403 {cpr 3528 |
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-nel 2404 df-v 2688 df-un 3075 df-sn 3533 df-pr 3534 |
This theorem is referenced by: (None) |
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