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Theorem ordtriexmid 4194
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 3205 . . . 4 ¬ {z {∅} ∣ φ}
2 ordtriexmidlem 4192 . . . . . 6 {z {∅} ∣ φ} On
3 eleq1 2082 . . . . . . . 8 (x = {z {∅} ∣ φ} → (x ∅ ↔ {z {∅} ∣ φ} ∅))
4 eqeq1 2028 . . . . . . . 8 (x = {z {∅} ∣ φ} → (x = ∅ ↔ {z {∅} ∣ φ} = ∅))
5 eleq2 2083 . . . . . . . 8 (x = {z {∅} ∣ φ} → (∅ x ↔ ∅ {z {∅} ∣ φ}))
63, 4, 53orbi123d 1191 . . . . . . 7 (x = {z {∅} ∣ φ} → ((x x = ∅ x) ↔ ({z {∅} ∣ φ} {z {∅} ∣ φ} = ∅ {z {∅} ∣ φ})))
7 0elon 4078 . . . . . . . 8 On
8 0ex 3858 . . . . . . . . 9 V
9 eleq1 2082 . . . . . . . . . . 11 (y = ∅ → (y On ↔ ∅ On))
109anbi2d 440 . . . . . . . . . 10 (y = ∅ → ((x On y On) ↔ (x On On)))
11 eleq2 2083 . . . . . . . . . . 11 (y = ∅ → (x yx ∅))
12 eqeq2 2031 . . . . . . . . . . 11 (y = ∅ → (x = yx = ∅))
13 eleq1 2082 . . . . . . . . . . 11 (y = ∅ → (y x ↔ ∅ x))
1411, 12, 133orbi123d 1191 . . . . . . . . . 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 2386 . . . . . . . . 9 ((x On y On) → (x y x = y y x))
188, 15, 17vtocl 2585 . . . . . . . 8 ((x On On) → (x x = ∅ x))
197, 18mpan2 403 . . . . . . 7 (x On → (x x = ∅ x))
206, 19vtoclga 2596 . . . . . 6 ({z {∅} ∣ φ} On → ({z {∅} ∣ φ} {z {∅} ∣ φ} = ∅ {z {∅} ∣ φ}))
212, 20ax-mp 7 . . . . 5 ({z {∅} ∣ φ} {z {∅} ∣ φ} = ∅ {z {∅} ∣ φ})
22 3orass 876 . . . . 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 1300 . . 3 ({z {∅} ∣ φ} = ∅ {z {∅} ∣ φ})
25 ordtriexmidlem2 4193 . . . 4 ({z {∅} ∣ φ} = ∅ → ¬ φ)
268snid 3377 . . . . . 6 {∅}
27 biidd 161 . . . . . . 7 (z = ∅ → (φφ))
2827elrab3 2676 . . . . . 6 (∅ {∅} → (∅ {z {∅} ∣ φ} ↔ φ))
2926, 28ax-mp 7 . . . . 5 (∅ {z {∅} ∣ φ} ↔ φ)
3029biimpi 113 . . . 4 (∅ {z {∅} ∣ φ} → φ)
3125, 30orim12i 663 . . 3 (({z {∅} ∣ φ} = ∅ {z {∅} ∣ φ}) → (¬ φ φ))
3224, 31ax-mp 7 . 2 φ φ)
33 orcom 634 . 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 616   w3o 872   = wceq 1228   wcel 1374  wral 2284  {crab 2288  c0 3201  {csn 3350  Oncon0 4049
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 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-nul 3857  ax-pow 3901
This theorem depends on definitions:  df-bi 110  df-3or 874  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-rab 2293  df-v 2537  df-dif 2897  df-un 2899  df-in 2901  df-ss 2908  df-nul 3202  df-pw 3336  df-sn 3356  df-uni 3555  df-tr 3829  df-iord 4052  df-on 4054  df-suc 4057
This theorem is referenced by: (None)
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