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Mirrors > Home > ILE Home > Th. List > axapti | GIF version |
Description: Apartness of reals is tight. Axiom for real and complex numbers, derived from set theory. (This restates ax-pre-apti 7460 with ordering on the extended reals.) (Contributed by Jim Kingdon, 29-Jan-2020.) |
Ref | Expression |
---|---|
axapti | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ¬ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴)) → 𝐴 = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltxrlt 7552 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ 𝐴 <ℝ 𝐵)) | |
2 | ltxrlt 7552 | . . . . . 6 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝐵 < 𝐴 ↔ 𝐵 <ℝ 𝐴)) | |
3 | 2 | ancoms 264 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐵 < 𝐴 ↔ 𝐵 <ℝ 𝐴)) |
4 | 1, 3 | orbi12d 742 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴 < 𝐵 ∨ 𝐵 < 𝐴) ↔ (𝐴 <ℝ 𝐵 ∨ 𝐵 <ℝ 𝐴))) |
5 | 4 | notbid 627 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (¬ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴) ↔ ¬ (𝐴 <ℝ 𝐵 ∨ 𝐵 <ℝ 𝐴))) |
6 | ax-pre-apti 7460 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ¬ (𝐴 <ℝ 𝐵 ∨ 𝐵 <ℝ 𝐴)) → 𝐴 = 𝐵) | |
7 | 6 | 3expia 1145 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (¬ (𝐴 <ℝ 𝐵 ∨ 𝐵 <ℝ 𝐴) → 𝐴 = 𝐵)) |
8 | 5, 7 | sylbid 148 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (¬ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴) → 𝐴 = 𝐵)) |
9 | 8 | 3impia 1140 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ¬ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴)) → 𝐴 = 𝐵) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 102 ↔ wb 103 ∨ wo 664 ∧ w3a 924 = wceq 1289 ∈ wcel 1438 class class class wbr 3845 ℝcr 7349 <ℝ cltrr 7354 < clt 7522 |
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 579 ax-in2 580 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-13 1449 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-sep 3957 ax-pow 4009 ax-pr 4036 ax-un 4260 ax-setind 4353 ax-cnex 7436 ax-resscn 7437 ax-pre-apti 7460 |
This theorem depends on definitions: df-bi 115 df-3an 926 df-tru 1292 df-fal 1295 df-nf 1395 df-sb 1693 df-eu 1951 df-mo 1952 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ne 2256 df-nel 2351 df-ral 2364 df-rex 2365 df-rab 2368 df-v 2621 df-dif 3001 df-un 3003 df-in 3005 df-ss 3012 df-pw 3431 df-sn 3452 df-pr 3453 df-op 3455 df-uni 3654 df-br 3846 df-opab 3900 df-xp 4444 df-pnf 7524 df-mnf 7525 df-ltxr 7527 |
This theorem is referenced by: lttri3 7565 reapti 8056 |
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