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Mirrors > Home > MPE Home > Th. List > nbrne1 | Structured version Visualization version GIF version |
Description: Two classes are different if they don't have the same relationship to a third class. (Contributed by NM, 3-Jun-2012.) |
Ref | Expression |
---|---|
nbrne1 | ⊢ ((𝐴𝑅𝐵 ∧ ¬ 𝐴𝑅𝐶) → 𝐵 ≠ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq2 4587 | . . . 4 ⊢ (𝐵 = 𝐶 → (𝐴𝑅𝐵 ↔ 𝐴𝑅𝐶)) | |
2 | 1 | biimpcd 238 | . . 3 ⊢ (𝐴𝑅𝐵 → (𝐵 = 𝐶 → 𝐴𝑅𝐶)) |
3 | 2 | necon3bd 2796 | . 2 ⊢ (𝐴𝑅𝐵 → (¬ 𝐴𝑅𝐶 → 𝐵 ≠ 𝐶)) |
4 | 3 | imp 444 | 1 ⊢ ((𝐴𝑅𝐵 ∧ ¬ 𝐴𝑅𝐶) → 𝐵 ≠ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 = wceq 1475 ≠ wne 2780 class class class wbr 4583 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 |
This theorem is referenced by: dalem43 34019 cdleme3h 34540 cdleme7ga 34553 cdlemeg46req 34835 cdlemh 35123 cdlemk12 35156 cdlemk12u 35178 lighneallem1 40060 |
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