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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 3111 | . . 3 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ↔ ¬ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶)) | |
4 | elpri 4591 | . . . 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 | syl2anc 586 | 1 ⊢ (𝜑 → ¬ 𝐴 ∈ {𝐵, 𝐶}) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 ∨ wo 843 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 {cpr 4571 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-v 3498 df-un 3943 df-sn 4570 df-pr 4572 |
This theorem is referenced by: renfdisj 10703 sumtp 15106 pmtrprfv3 18584 logbgcd1irr 25374 perfectlem2 25808 nbupgrres 27148 usgr2pthlem 27546 eupth2lem3lem6 28014 cycpmco2 30777 cyc2fvx 30778 mnuprdlem1 40615 mnuprdlem2 40616 perfectALTVlem2 43894 |
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