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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 2874 | . . 3 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ↔ ¬ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶)) | |
4 | elpri 4145 | . . . 4 ⊢ (𝐴 ∈ {𝐵, 𝐶} → (𝐴 = 𝐵 ∨ 𝐴 = 𝐶)) | |
5 | 4 | con3i 149 | . . 3 ⊢ (¬ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶) → ¬ 𝐴 ∈ {𝐵, 𝐶}) |
6 | 3, 5 | sylbi 206 | . 2 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) → ¬ 𝐴 ∈ {𝐵, 𝐶}) |
7 | 1, 2, 6 | syl2anc 691 | 1 ⊢ (𝜑 → ¬ 𝐴 ∈ {𝐵, 𝐶}) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 382 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 {cpr 4127 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-v 3175 df-un 3545 df-sn 4126 df-pr 4128 |
This theorem is referenced by: renfdisj 9977 sumtp 14322 pmtrprfv3 17697 perfectlem2 24755 perfectALTVlem2 40165 nbupgrres 40592 usgr2pthlem 40969 eupth2lem3lem6 41401 |
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