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| Mirrors > Home > MPE Home > Th. List > nelprd | 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, deduction version. (Contributed by Alexander van der Vekens, 25-Jan-2018.) |
| Ref | Expression |
|---|---|
| nelprd.1 | ⊢ (𝜑 → 𝐴 ≠ 𝐵) |
| nelprd.2 | ⊢ (𝜑 → 𝐴 ≠ 𝐶) |
| Ref | Expression |
|---|---|
| nelprd | ⊢ (𝜑 → ¬ 𝐴 ∈ {𝐵, 𝐶}) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | nelprd.1 | . 2 ⊢ (𝜑 → 𝐴 ≠ 𝐵) | |
| 2 | nelprd.2 | . 2 ⊢ (𝜑 → 𝐴 ≠ 𝐶) | |
| 3 | neanior 3035 | . . 3 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ↔ ¬ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶)) | |
| 4 | elpri 4649 | . . . 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 | syl2anc 584 | 1 ⊢ (𝜑 → ¬ 𝐴 ∈ {𝐵, 𝐶}) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∨ wo 848 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 {cpr 4628 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-v 3482 df-un 3956 df-sn 4627 df-pr 4629 |
| This theorem is referenced by: ord2eln012 8535 renfdisj 11321 sumtp 15785 pmtrprfv3 19472 logbgcd1irr 26837 perfectlem2 27274 nbupgrres 29381 usgr2pthlem 29783 eupth2lem3lem6 30252 cycpmco2 33153 cyc2fvx 33154 elrspunsn 33457 relogbzexpd 41976 dvrelog2b 42067 dvrelogpow2b 42069 aks4d1p1p4 42072 aks4d1p6 42082 aks6d1c7lem1 42181 mnuprdlem1 44291 mnuprdlem2 44292 perfectALTVlem2 47709 |
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