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| Mirrors > Home > MPE Home > Th. List > nelpri | Structured version Visualization version 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 3034 | . . 3 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ↔ ¬ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶)) | |
| 4 | elpri 4648 | . . . 4 ⊢ (𝐴 ∈ {𝐵, 𝐶} → (𝐴 = 𝐵 ∨ 𝐴 = 𝐶)) | |
| 5 | 4 | con3i 154 | . . 3 ⊢ (¬ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶) → ¬ 𝐴 ∈ {𝐵, 𝐶}) | 
| 6 | 3, 5 | sylbi 217 | . 2 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) → ¬ 𝐴 ∈ {𝐵, 𝐶}) | 
| 7 | 1, 2, 6 | mp2an 692 | 1 ⊢ ¬ 𝐴 ∈ {𝐵, 𝐶} | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 ∧ wa 395 ∨ wo 847 = wceq 1539 ∈ wcel 2107 ≠ wne 2939 {cpr 4627 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1542 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-v 3481 df-un 3955 df-sn 4626 df-pr 4628 | 
| This theorem is referenced by: prneli 4655 ex-dif 30443 ex-in 30445 ex-pss 30448 ex-res 30461 ex-hash 30473 | 
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