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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 4657 | . . . 4 ⊢ (𝐵 = 𝐶 → (𝐴𝑅𝐵 ↔ 𝐴𝑅𝐶)) | |
2 | 1 | biimpcd 239 | . . 3 ⊢ (𝐴𝑅𝐵 → (𝐵 = 𝐶 → 𝐴𝑅𝐶)) |
3 | 2 | necon3bd 2808 | . 2 ⊢ (𝐴𝑅𝐵 → (¬ 𝐴𝑅𝐶 → 𝐵 ≠ 𝐶)) |
4 | 3 | imp 445 | 1 ⊢ ((𝐴𝑅𝐵 ∧ ¬ 𝐴𝑅𝐶) → 𝐵 ≠ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 384 = wceq 1483 ≠ wne 2794 class class class wbr 4653 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 |
This theorem is referenced by: zeneo 15063 dalem43 35001 cdleme3h 35522 cdleme7ga 35535 cdlemeg46req 35817 cdlemh 36105 cdlemk12 36138 cdlemk12u 36160 lighneallem1 41522 |
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