Theorem tridceq 17011

Description: Real trichotomy implies decidability of real number equality. Or in other words, analytic LPO implies analytic WLPO (see trilpo 16997 and redcwlpo 17010). Thus, this is an analytic analogue to lpowlpo 7498. (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 8400	. . . . . . 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 723	. . . . . 6 ⊢ (𝑥 ≠ 𝑦 → (𝑥 = 𝑦 ∨ 𝑥 ≠ 𝑦))
5		necom 2504	. . . . . 6 ⊢ (𝑦 ≠ 𝑥 ↔ 𝑥 ≠ 𝑦)
6		dcne 2431	. . . . . 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 724	. . . . . 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 8400	. . . . . . 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 1345	. . 3 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) → DECID 𝑥 = 𝑦))
18	17	ralimdva 2617	. 2 ⊢ (𝑥 ∈ ℝ → (∀𝑦 ∈ ℝ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) → ∀𝑦 ∈ ℝ DECID 𝑥 = 𝑦))
19	18	ralimia 2611	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 720  DECID wdc 846   ∨ w3o 1008   ∈ wcel 2209   ≠ wne 2420  ∀wral 2528   class class class wbr 4125  ℝcr 8168   < clt 8350

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-pre-ltirr 8281

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-xp 4775  df-pnf 8352  df-mnf 8353  df-ltxr 8355

This theorem is referenced by:  dcapnconstALT  17017