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Mirrors > Home > ILE Home > Th. List > 3ne0 | GIF version |
Description: The number 3 is nonzero. (Contributed by FL, 17-Oct-2010.) (Proof shortened by Andrew Salmon, 7-May-2011.) |
Ref | Expression |
---|---|
3ne0 | ⊢ 3 ≠ 0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3re 9007 | . 2 ⊢ 3 ∈ ℝ | |
2 | 3pos 9027 | . 2 ⊢ 0 < 3 | |
3 | 1, 2 | gt0ne0ii 8458 | 1 ⊢ 3 ≠ 0 |
Colors of variables: wff set class |
Syntax hints: ≠ wne 2357 0cc0 7825 3c3 8985 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 ax-cnex 7916 ax-resscn 7917 ax-1cn 7918 ax-1re 7919 ax-icn 7920 ax-addcl 7921 ax-addrcl 7922 ax-mulcl 7923 ax-addcom 7925 ax-addass 7927 ax-i2m1 7930 ax-0lt1 7931 ax-0id 7933 ax-rnegex 7934 ax-pre-ltirr 7937 ax-pre-lttrn 7939 ax-pre-ltadd 7941 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-nel 2453 df-ral 2470 df-rex 2471 df-rab 2474 df-v 2751 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-br 4016 df-opab 4077 df-xp 4644 df-iota 5190 df-fv 5236 df-ov 5891 df-pnf 8008 df-mnf 8009 df-ltxr 8011 df-2 8992 df-3 8993 |
This theorem is referenced by: (None) |
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