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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 3109 | . . 3 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ↔ ¬ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶)) | |
4 | elpri 4582 | . . . 4 ⊢ (𝐴 ∈ {𝐵, 𝐶} → (𝐴 = 𝐵 ∨ 𝐴 = 𝐶)) | |
5 | 4 | con3i 157 | . . 3 ⊢ (¬ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶) → ¬ 𝐴 ∈ {𝐵, 𝐶}) |
6 | 3, 5 | sylbi 219 | . 2 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) → ¬ 𝐴 ∈ {𝐵, 𝐶}) |
7 | 1, 2, 6 | mp2an 690 | 1 ⊢ ¬ 𝐴 ∈ {𝐵, 𝐶} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 398 ∨ wo 843 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 {cpr 4562 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-v 3496 df-un 3940 df-sn 4561 df-pr 4563 |
This theorem is referenced by: prneli 4588 ex-dif 28196 ex-in 28198 ex-pss 28201 ex-res 28214 ex-hash 28226 |
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