negelrp - Intuitionistic Logic Explorer

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Theorem negelrp 10105

Description: Elementhood of a negation in the positive real numbers. (Contributed by Thierry Arnoux, 19-Sep-2018.)

**Assertion**
Ref	Expression
negelrp	⊢ (𝐴 ∈ ℝ → ( -𝐴 ∈ ℝ⁺ ↔ 𝐴 < 0))

**Proof of Theorem negelrp**
Step	Hyp	Ref	Expression
1		elrp 10073	. 2 ⊢ ( -𝐴 ∈ ℝ⁺ ↔ ( -𝐴 ∈ ℝ ∧ 0 < -𝐴))
2		lt0neg1 8803	. . 3 ⊢ (𝐴 ∈ ℝ → (𝐴 < 0 ↔ 0 < -𝐴))
3		renegcl 8594	. . . 4 ⊢ (𝐴 ∈ ℝ → -𝐴 ∈ ℝ)
4	3	biantrurd 305	. . 3 ⊢ (𝐴 ∈ ℝ → (0 < -𝐴 ↔ ( -𝐴 ∈ ℝ ∧ 0 < -𝐴)))
5	2, 4	bitr2d 189	. 2 ⊢ (𝐴 ∈ ℝ → (( -𝐴 ∈ ℝ ∧ 0 < -𝐴) ↔ 𝐴 < 0))
6	1, 5	bitrid 192	1 ⊢ (𝐴 ∈ ℝ → ( -𝐴 ∈ ℝ⁺ ↔ 𝐴 < 0))

Colors of variables: wff set class

This proof depends on syntax axioms: → wi 4 ∧ wa 104 ↔ wb 105 ∈ wcel 2209 class class class wbr 4131 ℝcr 8184 0cc0 8185 < clt 8366 -cneg 8505 ℝ⁺crp 10071

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-sep 4251 ax-pow 4313 ax-pr 4348 ax-un 4580 ax-setind 4686 ax-cnex 8276 ax-resscn 8277 ax-1cn 8278 ax-1re 8279 ax-icn 8280 ax-addcl 8281 ax-addrcl 8282 ax-mulcl 8283 ax-addcom 8285 ax-addass 8287 ax-distr 8289 ax-i2m1 8290 ax-0id 8293 ax-rnegex 8294 ax-cnre 8296 ax-pre-ltadd 8301

This proof depends on definitions: df-bi 117 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rab 2537 df-v 2823 df-sbc 3052 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-br 4132 df-opab 4194 df-id 4440 df-xp 4782 df-rel 4783 df-cnv 4784 df-co 4785 df-dm 4786 df-iota 5340 df-fun 5382 df-fv 5388 df-riota 6041 df-ov 6091 df-oprab 6092 df-mpo 6093 df-pnf 8368 df-mnf 8369 df-ltxr 8371 df-sub 8506 df-neg 8507 df-rp 10072

This theorem is used by: negelrpd 10106

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