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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 2292 | . . 3 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ↔ ¬ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶)) | |
4 | elpri 3398 | . . . 4 ⊢ (𝐴 ∈ {𝐵, 𝐶} → (𝐴 = 𝐵 ∨ 𝐴 = 𝐶)) | |
5 | 4 | con3i 562 | . . 3 ⊢ (¬ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶) → ¬ 𝐴 ∈ {𝐵, 𝐶}) |
6 | 3, 5 | sylbi 114 | . 2 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) → ¬ 𝐴 ∈ {𝐵, 𝐶}) |
7 | 1, 2, 6 | mp2an 402 | 1 ⊢ ¬ 𝐴 ∈ {𝐵, 𝐶} |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∧ wa 97 ∨ wo 629 = wceq 1243 ∈ wcel 1393 ≠ wne 2204 {cpr 3376 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-v 2559 df-un 2922 df-sn 3381 df-pr 3382 |
This theorem is referenced by: (None) |
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