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Mirrors > Home > NFE Home > Th. List > nelpri | 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 2601 | . . 3 ⊢ ((A ≠ B ∧ A ≠ C) ↔ ¬ (A = B ∨ A = C)) | |
4 | elpri 3753 | . . . 4 ⊢ (A ∈ {B, C} → (A = B ∨ A = C)) | |
5 | 4 | con3i 127 | . . 3 ⊢ (¬ (A = B ∨ A = C) → ¬ A ∈ {B, C}) |
6 | 3, 5 | sylbi 187 | . 2 ⊢ ((A ≠ B ∧ A ≠ C) → ¬ A ∈ {B, C}) |
7 | 1, 2, 6 | mp2an 653 | 1 ⊢ ¬ A ∈ {B, C} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∨ wo 357 ∧ wa 358 = wceq 1642 ∈ wcel 1710 ≠ wne 2516 {cpr 3738 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-v 2861 df-nin 3211 df-compl 3212 df-un 3214 df-sn 3741 df-pr 3742 |
This theorem is referenced by: (None) |
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