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Mirrors > Home > ILE Home > Th. List > subne0d | GIF version |
Description: Two unequal numbers have nonzero difference. See also subap0d 8599 which is the same thing for apartness rather than negated equality. (Contributed by Mario Carneiro, 1-Jan-2017.) |
Ref | Expression |
---|---|
negidd.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
pncand.2 | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
subne0d.3 | ⊢ (𝜑 → 𝐴 ≠ 𝐵) |
Ref | Expression |
---|---|
subne0d | ⊢ (𝜑 → (𝐴 − 𝐵) ≠ 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | subne0d.3 | . 2 ⊢ (𝜑 → 𝐴 ≠ 𝐵) | |
2 | negidd.1 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
3 | pncand.2 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
4 | subeq0 8181 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 − 𝐵) = 0 ↔ 𝐴 = 𝐵)) | |
5 | 2, 3, 4 | syl2anc 411 | . . 3 ⊢ (𝜑 → ((𝐴 − 𝐵) = 0 ↔ 𝐴 = 𝐵)) |
6 | 5 | necon3bid 2388 | . 2 ⊢ (𝜑 → ((𝐴 − 𝐵) ≠ 0 ↔ 𝐴 ≠ 𝐵)) |
7 | 1, 6 | mpbird 167 | 1 ⊢ (𝜑 → (𝐴 − 𝐵) ≠ 0) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 = wceq 1353 ∈ wcel 2148 ≠ wne 2347 (class class class)co 5874 ℂcc 7808 0cc0 7810 − cmin 8126 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 ax-setind 4536 ax-resscn 7902 ax-1cn 7903 ax-icn 7905 ax-addcl 7906 ax-addrcl 7907 ax-mulcl 7908 ax-addcom 7910 ax-addass 7912 ax-distr 7914 ax-i2m1 7915 ax-0id 7918 ax-rnegex 7919 ax-cnre 7921 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4004 df-opab 4065 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-iota 5178 df-fun 5218 df-fv 5224 df-riota 5830 df-ov 5877 df-oprab 5878 df-mpo 5879 df-sub 8128 |
This theorem is referenced by: modsumfzodifsn 10393 |
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