Theorem tridceq 16484

Description: Real trichotomy implies decidability of real number equality. Or in other words, analytic LPO implies analytic WLPO (see trilpo 16471 and redcwlpo 16483). Thus, this is an analytic analogue to lpowlpo 7346. (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 8242	. . . . . . 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 716	. . . . . 6 ⊢ (𝑥 ≠ 𝑦 → (𝑥 = 𝑦 ∨ 𝑥 ≠ 𝑦))
5		necom 2484	. . . . . 6 ⊢ (𝑦 ≠ 𝑥 ↔ 𝑥 ≠ 𝑦)
6		dcne 2411	. . . . . 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 717	. . . . . 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 8242	. . . . . . 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 1338	. . 3 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) → DECID 𝑥 = 𝑦))
18	17	ralimdva 2597	. 2 ⊢ (𝑥 ∈ ℝ → (∀𝑦 ∈ ℝ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) → ∀𝑦 ∈ ℝ DECID 𝑥 = 𝑦))
19	18	ralimia 2591	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 713  DECID wdc 839   ∨ w3o 1001   ∈ wcel 2200   ≠ wne 2400  ∀wral 2508   class class class wbr 4083  ℝcr 8009   < clt 8192

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8101  ax-resscn 8102  ax-pre-ltirr 8122

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-xp 4725  df-pnf 8194  df-mnf 8195  df-ltxr 8197

This theorem is referenced by:  dcapnconstALT  16490