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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 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 | mp2an 691 | 1 ⊢ ¬ 𝐴 ∈ {𝐵, 𝐶} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ 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: prneli 4621 ex-dif 29409 ex-in 29411 ex-pss 29414 ex-res 29427 ex-hash 29439 |
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