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Theorem tridceq 13638
Description: Real trichotomy implies decidability of real number equality. Or in other words, analytic LPO implies analytic WLPO (see trilpo 13625 and redcwlpo 13637). Thus, this is an analytic analogue to lpowlpo 7111. (Contributed by Jim Kingdon, 24-Jul-2024.)
Assertion
Ref Expression
tridceq (∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥) → ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ DECID 𝑥 = 𝑦)
Distinct variable group:   𝑥,𝑦

Proof of Theorem tridceq
StepHypRef Expression
1 ltne 7962 . . . . . . 7 ((𝑥 ∈ ℝ ∧ 𝑥 < 𝑦) → 𝑦𝑥)
21ex 114 . . . . . 6 (𝑥 ∈ ℝ → (𝑥 < 𝑦𝑦𝑥))
32adantr 274 . . . . 5 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 < 𝑦𝑦𝑥))
4 olc 701 . . . . . 6 (𝑥𝑦 → (𝑥 = 𝑦𝑥𝑦))
5 necom 2411 . . . . . 6 (𝑦𝑥𝑥𝑦)
6 dcne 2338 . . . . . 6 (DECID 𝑥 = 𝑦 ↔ (𝑥 = 𝑦𝑥𝑦))
74, 5, 63imtr4i 200 . . . . 5 (𝑦𝑥DECID 𝑥 = 𝑦)
83, 7syl6 33 . . . 4 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 < 𝑦DECID 𝑥 = 𝑦))
9 orc 702 . . . . . 6 (𝑥 = 𝑦 → (𝑥 = 𝑦𝑥𝑦))
109, 6sylibr 133 . . . . 5 (𝑥 = 𝑦DECID 𝑥 = 𝑦)
1110a1i 9 . . . 4 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 = 𝑦DECID 𝑥 = 𝑦))
12 ltne 7962 . . . . . . 7 ((𝑦 ∈ ℝ ∧ 𝑦 < 𝑥) → 𝑥𝑦)
1312ex 114 . . . . . 6 (𝑦 ∈ ℝ → (𝑦 < 𝑥𝑥𝑦))
1413adantl 275 . . . . 5 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑦 < 𝑥𝑥𝑦))
154, 6sylibr 133 . . . . 5 (𝑥𝑦DECID 𝑥 = 𝑦)
1614, 15syl6 33 . . . 4 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑦 < 𝑥DECID 𝑥 = 𝑦))
178, 11, 163jaod 1286 . . 3 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥) → DECID 𝑥 = 𝑦))
1817ralimdva 2524 . 2 (𝑥 ∈ ℝ → (∀𝑦 ∈ ℝ (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥) → ∀𝑦 ∈ ℝ DECID 𝑥 = 𝑦))
1918ralimia 2518 1 (∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥) → ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ DECID 𝑥 = 𝑦)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wo 698  DECID wdc 820  w3o 962  wcel 2128  wne 2327  wral 2435   class class class wbr 3965  cr 7731   < clt 7912
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-cnex 7823  ax-resscn 7824  ax-pre-ltirr 7844
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-xp 4592  df-pnf 7914  df-mnf 7915  df-ltxr 7917
This theorem is referenced by:  dcapnconstALT  13643
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