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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 5146 | . . . 4 ⊢ (𝐴 = 𝐵 → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐶)) | |
| 2 | 1 | biimpcd 249 | . . 3 ⊢ (𝐴𝑅𝐶 → (𝐴 = 𝐵 → 𝐵𝑅𝐶)) |
| 3 | 2 | necon3bd 2954 | . 2 ⊢ (𝐴𝑅𝐶 → (¬ 𝐵𝑅𝐶 → 𝐴 ≠ 𝐵)) |
| 4 | 3 | imp 406 | 1 ⊢ ((𝐴𝑅𝐶 ∧ ¬ 𝐵𝑅𝐶) → 𝐴 ≠ 𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 ≠ wne 2940 class class class wbr 5143 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 |
| This theorem is referenced by: frfi 9321 ablsimpgfindlem1 20127 ablsimpgfindlem2 20128 hl2at 39407 2atjm 39447 atbtwn 39448 atbtwnexOLDN 39449 atbtwnex 39450 dalem21 39696 dalem23 39698 dalem27 39701 dalem54 39728 2llnma1b 39788 lhpexle1lem 40009 lhpexle3lem 40013 lhp2at0nle 40037 4atexlemunv 40068 4atexlemnclw 40072 4atexlemcnd 40074 cdlemc5 40197 cdleme0b 40214 cdleme0c 40215 cdleme0fN 40220 cdleme01N 40223 cdleme0ex2N 40226 cdleme3b 40231 cdleme3c 40232 cdleme3g 40236 cdleme3h 40237 cdleme7aa 40244 cdleme7b 40246 cdleme7c 40247 cdleme7d 40248 cdleme7e 40249 cdleme7ga 40250 cdleme11fN 40266 cdlemesner 40298 cdlemednpq 40301 cdleme19a 40305 cdleme19c 40307 cdleme21c 40329 cdleme21ct 40331 cdleme22cN 40344 cdleme22f2 40349 cdleme22g 40350 cdleme41sn3aw 40476 cdlemeg46rgv 40530 cdlemeg46req 40531 cdlemf1 40563 cdlemg27b 40698 cdlemg33b0 40703 cdlemg33c0 40704 cdlemh 40819 cdlemk14 40856 dia2dimlem1 41066 |
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