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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 3038 | . . 3 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ↔ ¬ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶)) | |
4 | elpri 4613 | . . . 4 ⊢ (𝐴 ∈ {𝐵, 𝐶} → (𝐴 = 𝐵 ∨ 𝐴 = 𝐶)) | |
5 | 4 | con3i 154 | . . 3 ⊢ (¬ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶) → ¬ 𝐴 ∈ {𝐵, 𝐶}) |
6 | 3, 5 | sylbi 216 | . 2 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) → ¬ 𝐴 ∈ {𝐵, 𝐶}) |
7 | 1, 2, 6 | syl2anc 585 | 1 ⊢ (𝜑 → ¬ 𝐴 ∈ {𝐵, 𝐶}) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 397 ∨ wo 846 = wceq 1542 ∈ wcel 2107 ≠ wne 2944 {cpr 4593 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-ne 2945 df-v 3450 df-un 3920 df-sn 4592 df-pr 4594 |
This theorem is referenced by: ord2eln012 8448 renfdisj 11222 sumtp 15641 pmtrprfv3 19243 logbgcd1irr 26160 perfectlem2 26594 nbupgrres 28354 usgr2pthlem 28753 eupth2lem3lem6 29219 cycpmco2 32024 cyc2fvx 32025 relogbzexpd 40461 dvrelog2b 40552 dvrelogpow2b 40554 aks4d1p1p4 40557 aks4d1p6 40567 mnuprdlem1 42626 mnuprdlem2 42627 perfectALTVlem2 45988 |
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