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Mirrors > Home > MPE Home > Th. List > nbrne2 | 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 |
---|---|
nbrne2 | ⊢ ((𝐴𝑅𝐶 ∧ ¬ 𝐵𝑅𝐶) → 𝐴 ≠ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq1 5152 | . . . 4 ⊢ (𝐴 = 𝐵 → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐶)) | |
2 | 1 | biimpcd 248 | . . 3 ⊢ (𝐴𝑅𝐶 → (𝐴 = 𝐵 → 𝐵𝑅𝐶)) |
3 | 2 | necon3bd 2943 | . 2 ⊢ (𝐴𝑅𝐶 → (¬ 𝐵𝑅𝐶 → 𝐴 ≠ 𝐵)) |
4 | 3 | imp 405 | 1 ⊢ ((𝐴𝑅𝐶 ∧ ¬ 𝐵𝑅𝐶) → 𝐴 ≠ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 394 = wceq 1533 ≠ wne 2929 class class class wbr 5149 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-ne 2930 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5150 |
This theorem is referenced by: frfi 9316 ablsimpgfindlem1 20081 ablsimpgfindlem2 20082 hl2at 39010 2atjm 39050 atbtwn 39051 atbtwnexOLDN 39052 atbtwnex 39053 dalem21 39299 dalem23 39301 dalem27 39304 dalem54 39331 2llnma1b 39391 lhpexle1lem 39612 lhpexle3lem 39616 lhp2at0nle 39640 4atexlemunv 39671 4atexlemnclw 39675 4atexlemcnd 39677 cdlemc5 39800 cdleme0b 39817 cdleme0c 39818 cdleme0fN 39823 cdleme01N 39826 cdleme0ex2N 39829 cdleme3b 39834 cdleme3c 39835 cdleme3g 39839 cdleme3h 39840 cdleme7aa 39847 cdleme7b 39849 cdleme7c 39850 cdleme7d 39851 cdleme7e 39852 cdleme7ga 39853 cdleme11fN 39869 cdlemesner 39901 cdlemednpq 39904 cdleme19a 39908 cdleme19c 39910 cdleme21c 39932 cdleme21ct 39934 cdleme22cN 39947 cdleme22f2 39952 cdleme22g 39953 cdleme41sn3aw 40079 cdlemeg46rgv 40133 cdlemeg46req 40134 cdlemf1 40166 cdlemg27b 40301 cdlemg33b0 40306 cdlemg33c0 40307 cdlemh 40422 cdlemk14 40459 dia2dimlem1 40669 |
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