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Mirrors > Home > ILE Home > Th. List > Mathboxes > tridceq | GIF version |
Description: Real trichotomy implies decidability of real number equality. Or in other words, analytic LPO implies analytic WLPO (see trilpo 14830 and redcwlpo 14842). Thus, this is an analytic analogue to lpowlpo 7168. (Contributed by Jim Kingdon, 24-Jul-2024.) |
Ref | Expression |
---|---|
tridceq | ⊢ (∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) → ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ DECID 𝑥 = 𝑦) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltne 8044 | . . . . . . 7 ⊢ ((𝑥 ∈ ℝ ∧ 𝑥 < 𝑦) → 𝑦 ≠ 𝑥) | |
2 | 1 | ex 115 | . . . . . 6 ⊢ (𝑥 ∈ ℝ → (𝑥 < 𝑦 → 𝑦 ≠ 𝑥)) |
3 | 2 | adantr 276 | . . . . 5 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 < 𝑦 → 𝑦 ≠ 𝑥)) |
4 | olc 711 | . . . . . 6 ⊢ (𝑥 ≠ 𝑦 → (𝑥 = 𝑦 ∨ 𝑥 ≠ 𝑦)) | |
5 | necom 2431 | . . . . . 6 ⊢ (𝑦 ≠ 𝑥 ↔ 𝑥 ≠ 𝑦) | |
6 | dcne 2358 | . . . . . 6 ⊢ (DECID 𝑥 = 𝑦 ↔ (𝑥 = 𝑦 ∨ 𝑥 ≠ 𝑦)) | |
7 | 4, 5, 6 | 3imtr4i 201 | . . . . 5 ⊢ (𝑦 ≠ 𝑥 → DECID 𝑥 = 𝑦) |
8 | 3, 7 | syl6 33 | . . . 4 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 < 𝑦 → DECID 𝑥 = 𝑦)) |
9 | orc 712 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑦 ∨ 𝑥 ≠ 𝑦)) | |
10 | 9, 6 | sylibr 134 | . . . . 5 ⊢ (𝑥 = 𝑦 → DECID 𝑥 = 𝑦) |
11 | 10 | a1i 9 | . . . 4 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 = 𝑦 → DECID 𝑥 = 𝑦)) |
12 | ltne 8044 | . . . . . . 7 ⊢ ((𝑦 ∈ ℝ ∧ 𝑦 < 𝑥) → 𝑥 ≠ 𝑦) | |
13 | 12 | ex 115 | . . . . . 6 ⊢ (𝑦 ∈ ℝ → (𝑦 < 𝑥 → 𝑥 ≠ 𝑦)) |
14 | 13 | adantl 277 | . . . . 5 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑦 < 𝑥 → 𝑥 ≠ 𝑦)) |
15 | 4, 6 | sylibr 134 | . . . . 5 ⊢ (𝑥 ≠ 𝑦 → DECID 𝑥 = 𝑦) |
16 | 14, 15 | syl6 33 | . . . 4 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑦 < 𝑥 → DECID 𝑥 = 𝑦)) |
17 | 8, 11, 16 | 3jaod 1304 | . . 3 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) → DECID 𝑥 = 𝑦)) |
18 | 17 | ralimdva 2544 | . 2 ⊢ (𝑥 ∈ ℝ → (∀𝑦 ∈ ℝ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) → ∀𝑦 ∈ ℝ DECID 𝑥 = 𝑦)) |
19 | 18 | ralimia 2538 | 1 ⊢ (∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) → ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ DECID 𝑥 = 𝑦) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∨ wo 708 DECID wdc 834 ∨ w3o 977 ∈ wcel 2148 ≠ wne 2347 ∀wral 2455 class class class wbr 4005 ℝcr 7812 < clt 7994 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4123 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-cnex 7904 ax-resscn 7905 ax-pre-ltirr 7925 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2741 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-br 4006 df-opab 4067 df-xp 4634 df-pnf 7996 df-mnf 7997 df-ltxr 7999 |
This theorem is referenced by: dcapnconstALT 14849 |
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