Theorem tridceq 16841

Description: Real trichotomy implies decidability of real number equality. Or in other words, analytic LPO implies analytic WLPO (see trilpo 16827 and redcwlpo 16840). Thus, this is an analytic analogue to lpowlpo 7459. (Contributed by Jim Kingdon, 24-Jul-2024.)

Assertion

Ref	Expression
tridceq	⊢ (∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) → ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ DECID 𝑥 = 𝑦)

Distinct variable group:   𝑥,𝑦

Proof of Theorem tridceq

Step	Hyp	Ref	Expression
1		ltne 8358	. . . . . . 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 719	. . . . . 6 ⊢ (𝑥 ≠ 𝑦 → (𝑥 = 𝑦 ∨ 𝑥 ≠ 𝑦))
5		necom 2496	. . . . . 6 ⊢ (𝑦 ≠ 𝑥 ↔ 𝑥 ≠ 𝑦)
6		dcne 2423	. . . . . 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 720	. . . . . 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 8358	. . . . . . 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 1341	. . 3 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) → DECID 𝑥 = 𝑦))
18	17	ralimdva 2609	. 2 ⊢ (𝑥 ∈ ℝ → (∀𝑦 ∈ ℝ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) → ∀𝑦 ∈ ℝ DECID 𝑥 = 𝑦))
19	18	ralimia 2603	1 ⊢ (∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) → ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ DECID 𝑥 = 𝑦)

Syntax hints:   → wi 4   ∧ wa 104   ∨ wo 716  DECID wdc 842   ∨ w3o 1004   ∈ wcel 2203   ≠ wne 2412  ∀wral 2520   class class class wbr 4109  ℝcr 8126   < clt 8308

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-pre-ltirr 8239

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2815  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-xp 4755  df-pnf 8310  df-mnf 8311  df-ltxr 8313

This theorem is referenced by:  dcapnconstALT  16848