Users' Mathboxes Mathbox for Jim Kingdon < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >   Mathboxes  >  tridceq GIF version

Theorem tridceq 14088
Description: Real trichotomy implies decidability of real number equality. Or in other words, analytic LPO implies analytic WLPO (see trilpo 14075 and redcwlpo 14087). Thus, this is an analytic analogue to lpowlpo 7144. (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 8004 . . . . . . 7 ((𝑥 ∈ ℝ ∧ 𝑥 < 𝑦) → 𝑦𝑥)
21ex 114 . . . . . 6 (𝑥 ∈ ℝ → (𝑥 < 𝑦𝑦𝑥))
32adantr 274 . . . . 5 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 < 𝑦𝑦𝑥))
4 olc 706 . . . . . 6 (𝑥𝑦 → (𝑥 = 𝑦𝑥𝑦))
5 necom 2424 . . . . . 6 (𝑦𝑥𝑥𝑦)
6 dcne 2351 . . . . . 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 707 . . . . . 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 8004 . . . . . . 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 1299 . . 3 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥) → DECID 𝑥 = 𝑦))
1817ralimdva 2537 . 2 (𝑥 ∈ ℝ → (∀𝑦 ∈ ℝ (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥) → ∀𝑦 ∈ ℝ DECID 𝑥 = 𝑦))
1918ralimia 2531 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 703  DECID wdc 829  w3o 972  wcel 2141  wne 2340  wral 2448   class class class wbr 3989  cr 7773   < clt 7954
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-pre-ltirr 7886
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-xp 4617  df-pnf 7956  df-mnf 7957  df-ltxr 7959
This theorem is referenced by:  dcapnconstALT  14093
  Copyright terms: Public domain W3C validator