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Theorem tridceq 14664
Description: Real trichotomy implies decidability of real number equality. Or in other words, analytic LPO implies analytic WLPO (see trilpo 14651 and redcwlpo 14663). Thus, this is an analytic analogue to lpowlpo 7163. (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 8038 . . . . . . 7 ((𝑥 ∈ ℝ ∧ 𝑥 < 𝑦) → 𝑦𝑥)
21ex 115 . . . . . 6 (𝑥 ∈ ℝ → (𝑥 < 𝑦𝑦𝑥))
32adantr 276 . . . . 5 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 < 𝑦𝑦𝑥))
4 olc 711 . . . . . 6 (𝑥𝑦 → (𝑥 = 𝑦𝑥𝑦))
5 necom 2431 . . . . . 6 (𝑦𝑥𝑥𝑦)
6 dcne 2358 . . . . . 6 (DECID 𝑥 = 𝑦 ↔ (𝑥 = 𝑦𝑥𝑦))
74, 5, 63imtr4i 201 . . . . 5 (𝑦𝑥DECID 𝑥 = 𝑦)
83, 7syl6 33 . . . 4 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 < 𝑦DECID 𝑥 = 𝑦))
9 orc 712 . . . . . 6 (𝑥 = 𝑦 → (𝑥 = 𝑦𝑥𝑦))
109, 6sylibr 134 . . . . 5 (𝑥 = 𝑦DECID 𝑥 = 𝑦)
1110a1i 9 . . . 4 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 = 𝑦DECID 𝑥 = 𝑦))
12 ltne 8038 . . . . . . 7 ((𝑦 ∈ ℝ ∧ 𝑦 < 𝑥) → 𝑥𝑦)
1312ex 115 . . . . . 6 (𝑦 ∈ ℝ → (𝑦 < 𝑥𝑥𝑦))
1413adantl 277 . . . . 5 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑦 < 𝑥𝑥𝑦))
154, 6sylibr 134 . . . . 5 (𝑥𝑦DECID 𝑥 = 𝑦)
1614, 15syl6 33 . . . 4 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑦 < 𝑥DECID 𝑥 = 𝑦))
178, 11, 163jaod 1304 . . 3 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥) → DECID 𝑥 = 𝑦))
1817ralimdva 2544 . 2 (𝑥 ∈ ℝ → (∀𝑦 ∈ ℝ (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥) → ∀𝑦 ∈ ℝ DECID 𝑥 = 𝑦))
1918ralimia 2538 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 104  wo 708  DECID wdc 834  w3o 977  wcel 2148  wne 2347  wral 2455   class class class wbr 4002  cr 7807   < clt 7988
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4120  ax-pow 4173  ax-pr 4208  ax-un 4432  ax-setind 4535  ax-cnex 7899  ax-resscn 7900  ax-pre-ltirr 7920
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4003  df-opab 4064  df-xp 4631  df-pnf 7990  df-mnf 7991  df-ltxr 7993
This theorem is referenced by:  dcapnconstALT  14669
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