![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > posdifd | GIF version |
Description: Comparison of two numbers whose difference is positive. (Contributed by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
leidd.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
ltnegd.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
Ref | Expression |
---|---|
posdifd | ⊢ (𝜑 → (𝐴 < 𝐵 ↔ 0 < (𝐵 − 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | leidd.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
2 | ltnegd.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
3 | posdif 8442 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ 0 < (𝐵 − 𝐴))) | |
4 | 1, 2, 3 | syl2anc 411 | 1 ⊢ (𝜑 → (𝐴 < 𝐵 ↔ 0 < (𝐵 − 𝐴))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 ∈ wcel 2160 class class class wbr 4018 (class class class)co 5896 ℝcr 7840 0cc0 7841 < clt 8022 − cmin 8158 |
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-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-cnex 7932 ax-resscn 7933 ax-1cn 7934 ax-1re 7935 ax-icn 7936 ax-addcl 7937 ax-addrcl 7938 ax-mulcl 7939 ax-addcom 7941 ax-addass 7943 ax-distr 7945 ax-i2m1 7946 ax-0id 7949 ax-rnegex 7950 ax-cnre 7952 ax-pre-ltadd 7957 |
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-nel 2456 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 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-iota 5196 df-fun 5237 df-fv 5243 df-riota 5852 df-ov 5899 df-oprab 5900 df-mpo 5901 df-pnf 8024 df-mnf 8025 df-ltxr 8027 df-sub 8160 df-neg 8161 |
This theorem is referenced by: possumd 8556 ltmul1a 8578 ztri3or 9326 qbtwnre 10287 expnbnd 10675 resqrexlemover 11051 fsumlt 11504 cvgratnnlembern 11563 cvgratnnlemsumlt 11568 cvgratnnlemfm 11569 cvgratnnlemrate 11570 cvgratnn 11571 sin01gt0 11801 cos12dec 11807 nno 11943 pythagtriplem10 12301 ivthinclemlopn 14571 ivthinclemuopn 14573 sin0pilem1 14659 cosordlem 14727 cosq34lt1 14728 iooref1o 15241 trilpolemlt1 15248 trirec0 15251 |
Copyright terms: Public domain | W3C validator |