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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 3091 | . . 3 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ↔ ¬ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶)) | |
4 | elpri 4419 | . . . 4 ⊢ (𝐴 ∈ {𝐵, 𝐶} → (𝐴 = 𝐵 ∨ 𝐴 = 𝐶)) | |
5 | 4 | con3i 152 | . . 3 ⊢ (¬ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶) → ¬ 𝐴 ∈ {𝐵, 𝐶}) |
6 | 3, 5 | sylbi 209 | . 2 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) → ¬ 𝐴 ∈ {𝐵, 𝐶}) |
7 | 1, 2, 6 | syl2anc 581 | 1 ⊢ (𝜑 → ¬ 𝐴 ∈ {𝐵, 𝐶}) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 386 ∨ wo 880 = wceq 1658 ∈ wcel 2166 ≠ wne 2999 {cpr 4399 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-ext 2803 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-v 3416 df-un 3803 df-sn 4398 df-pr 4400 |
This theorem is referenced by: renfdisj 10417 sumtp 14855 pmtrprfv3 18224 logbgcd1irr 24934 perfectlem2 25368 nbupgrres 26661 usgr2pthlem 27065 eupth2lem3lem6 27610 perfectALTVlem2 42461 |
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