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Mirrors > Home > ILE Home > Th. List > negrebi | GIF version |
Description: The negative of a real is real. (Contributed by NM, 11-Aug-1999.) |
Ref | Expression |
---|---|
negidi.1 | ⊢ 𝐴 ∈ ℂ |
Ref | Expression |
---|---|
negrebi | ⊢ (-𝐴 ∈ ℝ ↔ 𝐴 ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | negidi.1 | . 2 ⊢ 𝐴 ∈ ℂ | |
2 | negreb 8270 | . 2 ⊢ (𝐴 ∈ ℂ → (-𝐴 ∈ ℝ ↔ 𝐴 ∈ ℝ)) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (-𝐴 ∈ ℝ ↔ 𝐴 ∈ ℝ) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 105 ∈ wcel 2160 ℂcc 7856 ℝcr 7857 -cneg 8177 |
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 4143 ax-pow 4199 ax-pr 4234 ax-setind 4561 ax-resscn 7950 ax-1cn 7951 ax-icn 7953 ax-addcl 7954 ax-addrcl 7955 ax-mulcl 7956 ax-addcom 7958 ax-addass 7960 ax-distr 7962 ax-i2m1 7963 ax-0id 7966 ax-rnegex 7967 ax-cnre 7969 |
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 2758 df-sbc 2982 df-dif 3151 df-un 3153 df-in 3155 df-ss 3162 df-pw 3599 df-sn 3620 df-pr 3621 df-op 3623 df-uni 3832 df-br 4026 df-opab 4087 df-id 4318 df-xp 4657 df-rel 4658 df-cnv 4659 df-co 4660 df-dm 4661 df-iota 5203 df-fun 5244 df-fv 5250 df-riota 5861 df-ov 5909 df-oprab 5910 df-mpo 5911 df-sub 8178 df-neg 8179 |
This theorem is referenced by: (None) |
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