Theorem tridceq 15928

Description: Real trichotomy implies decidability of real number equality. Or in other words, analytic LPO implies analytic WLPO (see trilpo 15915 and redcwlpo 15927). Thus, this is an analytic analogue to lpowlpo 7269. (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 8156	. . . . . . 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 712	. . . . . 6 ⊢ (𝑥 ≠ 𝑦 → (𝑥 = 𝑦 ∨ 𝑥 ≠ 𝑦))
5		necom 2459	. . . . . 6 ⊢ (𝑦 ≠ 𝑥 ↔ 𝑥 ≠ 𝑦)
6		dcne 2386	. . . . . 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 713	. . . . . 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 8156	. . . . . . 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 1316	. . 3 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) → DECID 𝑥 = 𝑦))
18	17	ralimdva 2572	. 2 ⊢ (𝑥 ∈ ℝ → (∀𝑦 ∈ ℝ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) → ∀𝑦 ∈ ℝ DECID 𝑥 = 𝑦))
19	18	ralimia 2566	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 709  DECID wdc 835   ∨ w3o 979   ∈ wcel 2175   ≠ wne 2375  ∀wral 2483   class class class wbr 4043  ℝcr 7923   < clt 8106

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 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584  ax-cnex 8015  ax-resscn 8016  ax-pre-ltirr 8036

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-nel 2471  df-ral 2488  df-rex 2489  df-rab 2492  df-v 2773  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-br 4044  df-opab 4105  df-xp 4680  df-pnf 8108  df-mnf 8109  df-ltxr 8111

This theorem is referenced by:  dcapnconstALT  15934