ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ordtriexmid Structured version   GIF version

Theorem ordtriexmid 4210
Description: Ordinal trichotomy implies the law of the excluded middle (that is, decidability of an arbitrary proposition).

This theorem is stated in "Constructive ordinals", [Crosilla], p. "Set-theoretic principles incompatible with intuitionistic logic".

(Contributed by Mario Carneiro and Jim Kingdon, 14-Nov-2018.)

Hypothesis
Ref Expression
ordtriexmid.1 x On y On (x y x = y y x)
Assertion
Ref Expression
ordtriexmid (φ ¬ φ)
Distinct variable groups:   x,y   φ,x
Allowed substitution hint:   φ(y)

Proof of Theorem ordtriexmid
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 noel 3222 . . . 4 ¬ {z {∅} ∣ φ}
2 ordtriexmidlem 4208 . . . . . 6 {z {∅} ∣ φ} On
3 eleq1 2097 . . . . . . . 8 (x = {z {∅} ∣ φ} → (x ∅ ↔ {z {∅} ∣ φ} ∅))
4 eqeq1 2043 . . . . . . . 8 (x = {z {∅} ∣ φ} → (x = ∅ ↔ {z {∅} ∣ φ} = ∅))
5 eleq2 2098 . . . . . . . 8 (x = {z {∅} ∣ φ} → (∅ x ↔ ∅ {z {∅} ∣ φ}))
63, 4, 53orbi123d 1205 . . . . . . 7 (x = {z {∅} ∣ φ} → ((x x = ∅ x) ↔ ({z {∅} ∣ φ} {z {∅} ∣ φ} = ∅ {z {∅} ∣ φ})))
7 0elon 4095 . . . . . . . 8 On
8 0ex 3875 . . . . . . . . 9 V
9 eleq1 2097 . . . . . . . . . . 11 (y = ∅ → (y On ↔ ∅ On))
109anbi2d 437 . . . . . . . . . 10 (y = ∅ → ((x On y On) ↔ (x On On)))
11 eleq2 2098 . . . . . . . . . . 11 (y = ∅ → (x yx ∅))
12 eqeq2 2046 . . . . . . . . . . 11 (y = ∅ → (x = yx = ∅))
13 eleq1 2097 . . . . . . . . . . 11 (y = ∅ → (y x ↔ ∅ x))
1411, 12, 133orbi123d 1205 . . . . . . . . . 10 (y = ∅ → ((x y x = y y x) ↔ (x x = ∅ x)))
1510, 14imbi12d 223 . . . . . . . . 9 (y = ∅ → (((x On y On) → (x y x = y y x)) ↔ ((x On On) → (x x = ∅ x))))
16 ordtriexmid.1 . . . . . . . . . 10 x On y On (x y x = y y x)
1716rspec2 2402 . . . . . . . . 9 ((x On y On) → (x y x = y y x))
188, 15, 17vtocl 2602 . . . . . . . 8 ((x On On) → (x x = ∅ x))
197, 18mpan2 401 . . . . . . 7 (x On → (x x = ∅ x))
206, 19vtoclga 2613 . . . . . 6 ({z {∅} ∣ φ} On → ({z {∅} ∣ φ} {z {∅} ∣ φ} = ∅ {z {∅} ∣ φ}))
212, 20ax-mp 7 . . . . 5 ({z {∅} ∣ φ} {z {∅} ∣ φ} = ∅ {z {∅} ∣ φ})
22 3orass 887 . . . . 5 (({z {∅} ∣ φ} {z {∅} ∣ φ} = ∅ {z {∅} ∣ φ}) ↔ ({z {∅} ∣ φ} ({z {∅} ∣ φ} = ∅ {z {∅} ∣ φ})))
2321, 22mpbi 133 . . . 4 ({z {∅} ∣ φ} ({z {∅} ∣ φ} = ∅ {z {∅} ∣ φ}))
241, 23mtp-or 1314 . . 3 ({z {∅} ∣ φ} = ∅ {z {∅} ∣ φ})
25 ordtriexmidlem2 4209 . . . 4 ({z {∅} ∣ φ} = ∅ → ¬ φ)
268snid 3394 . . . . . 6 {∅}
27 biidd 161 . . . . . . 7 (z = ∅ → (φφ))
2827elrab3 2693 . . . . . 6 (∅ {∅} → (∅ {z {∅} ∣ φ} ↔ φ))
2926, 28ax-mp 7 . . . . 5 (∅ {z {∅} ∣ φ} ↔ φ)
3029biimpi 113 . . . 4 (∅ {z {∅} ∣ φ} → φ)
3125, 30orim12i 675 . . 3 (({z {∅} ∣ φ} = ∅ {z {∅} ∣ φ}) → (¬ φ φ))
3224, 31ax-mp 7 . 2 φ φ)
33 orcom 646 . 2 ((φ ¬ φ) ↔ (¬ φ φ))
3432, 33mpbir 134 1 (φ ¬ φ)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97  wb 98   wo 628   w3o 883   = wceq 1242   wcel 1390  wral 2300  {crab 2304  c0 3218  {csn 3367  Oncon0 4066
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-nul 3874  ax-pow 3918
This theorem depends on definitions:  df-bi 110  df-3or 885  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-uni 3572  df-tr 3846  df-iord 4069  df-on 4071  df-suc 4074
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator