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Mirrors > Home > ILE Home > Th. List > ltleap | GIF version |
Description: Less than in terms of non-strict order and apartness. (Contributed by Jim Kingdon, 28-Feb-2020.) |
Ref | Expression |
---|---|
ltleap | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ (𝐴 ≤ 𝐵 ∧ 𝐴 # 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltle 7474 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 → 𝐴 ≤ 𝐵)) | |
2 | orc 666 | . . . 4 ⊢ (𝐴 < 𝐵 → (𝐴 < 𝐵 ∨ 𝐵 < 𝐴)) | |
3 | reaplt 7964 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 # 𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴))) | |
4 | 2, 3 | syl5ibr 154 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 → 𝐴 # 𝐵)) |
5 | 1, 4 | jcad 301 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 → (𝐴 ≤ 𝐵 ∧ 𝐴 # 𝐵))) |
6 | simprl 498 | . . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐴 ≤ 𝐵 ∧ 𝐴 # 𝐵)) → 𝐴 ≤ 𝐵) | |
7 | lenlt 7463 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴)) | |
8 | 7 | adantr 270 | . . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐴 ≤ 𝐵 ∧ 𝐴 # 𝐵)) → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴)) |
9 | 6, 8 | mpbid 145 | . . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐴 ≤ 𝐵 ∧ 𝐴 # 𝐵)) → ¬ 𝐵 < 𝐴) |
10 | simprr 499 | . . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐴 ≤ 𝐵 ∧ 𝐴 # 𝐵)) → 𝐴 # 𝐵) | |
11 | 3 | adantr 270 | . . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐴 ≤ 𝐵 ∧ 𝐴 # 𝐵)) → (𝐴 # 𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴))) |
12 | 10, 11 | mpbid 145 | . . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐴 ≤ 𝐵 ∧ 𝐴 # 𝐵)) → (𝐴 < 𝐵 ∨ 𝐵 < 𝐴)) |
13 | 9, 12 | ecased 1281 | . . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐴 ≤ 𝐵 ∧ 𝐴 # 𝐵)) → 𝐴 < 𝐵) |
14 | 13 | ex 113 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴 ≤ 𝐵 ∧ 𝐴 # 𝐵) → 𝐴 < 𝐵)) |
15 | 5, 14 | impbid 127 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ (𝐴 ≤ 𝐵 ∧ 𝐴 # 𝐵))) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 102 ↔ wb 103 ∨ wo 662 ∈ wcel 1434 class class class wbr 3811 ℝcr 7251 < clt 7424 ≤ cle 7425 # cap 7957 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-sep 3922 ax-pow 3974 ax-pr 3999 ax-un 4223 ax-setind 4315 ax-cnex 7338 ax-resscn 7339 ax-1cn 7340 ax-1re 7341 ax-icn 7342 ax-addcl 7343 ax-addrcl 7344 ax-mulcl 7345 ax-mulrcl 7346 ax-addcom 7347 ax-mulcom 7348 ax-addass 7349 ax-mulass 7350 ax-distr 7351 ax-i2m1 7352 ax-0lt1 7353 ax-1rid 7354 ax-0id 7355 ax-rnegex 7356 ax-precex 7357 ax-cnre 7358 ax-pre-ltirr 7359 ax-pre-lttrn 7361 ax-pre-apti 7362 ax-pre-ltadd 7363 ax-pre-mulgt0 7364 |
This theorem depends on definitions: df-bi 115 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ne 2250 df-nel 2345 df-ral 2358 df-rex 2359 df-reu 2360 df-rab 2362 df-v 2614 df-sbc 2827 df-dif 2986 df-un 2988 df-in 2990 df-ss 2997 df-pw 3408 df-sn 3428 df-pr 3429 df-op 3431 df-uni 3628 df-br 3812 df-opab 3866 df-id 4083 df-xp 4406 df-rel 4407 df-cnv 4408 df-co 4409 df-dm 4410 df-iota 4933 df-fun 4970 df-fv 4976 df-riota 5546 df-ov 5593 df-oprab 5594 df-mpt2 5595 df-pnf 7426 df-mnf 7427 df-xr 7428 df-ltxr 7429 df-le 7430 df-sub 7557 df-neg 7558 df-reap 7951 df-ap 7958 |
This theorem is referenced by: recgt0 8204 prodgt0 8206 lt2msq 8240 zltlen 8720 qltlen 9019 |
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