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Mirrors > Home > ILE Home > Th. List > subne0ad | Unicode version |
Description: If the difference of two complex numbers is nonzero, they are unequal. Converse of subne0d 8296. Contrapositive of subeq0bd 8355. (Contributed by David Moews, 28-Feb-2017.) |
Ref | Expression |
---|---|
negidd.1 |
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pncand.2 |
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subne0ad.3 |
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Ref | Expression |
---|---|
subne0ad |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | subne0ad.3 |
. 2
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2 | negidd.1 |
. . . 4
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3 | pncand.2 |
. . . 4
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4 | 2, 3 | subeq0ad 8297 |
. . 3
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5 | 4 | necon3bid 2401 |
. 2
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6 | 1, 5 | mpbid 147 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() |
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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4189 ax-pr 4224 ax-setind 4551 ax-resscn 7922 ax-1cn 7923 ax-icn 7925 ax-addcl 7926 ax-addrcl 7927 ax-mulcl 7928 ax-addcom 7930 ax-addass 7932 ax-distr 7934 ax-i2m1 7935 ax-0id 7938 ax-rnegex 7939 ax-cnre 7941 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-id 4308 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-iota 5193 df-fun 5233 df-fv 5239 df-riota 5847 df-ov 5894 df-oprab 5895 df-mpo 5896 df-sub 8149 |
This theorem is referenced by: (None) |
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