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Theorem ordtriexmid 4479
 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
Assertion
Ref Expression
ordtriexmid
Distinct variable groups:   ,   ,
Allowed substitution hint:   ()

Proof of Theorem ordtriexmid
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 noel 3398 . . . 4
2 ordtriexmidlem 4477 . . . . . 6
3 eleq1 2220 . . . . . . . 8
4 eqeq1 2164 . . . . . . . 8
5 eleq2 2221 . . . . . . . 8
63, 4, 53orbi123d 1293 . . . . . . 7
7 0elon 4352 . . . . . . . 8
8 0ex 4091 . . . . . . . . 9
9 eleq1 2220 . . . . . . . . . . 11
109anbi2d 460 . . . . . . . . . 10
11 eleq2 2221 . . . . . . . . . . 11
12 eqeq2 2167 . . . . . . . . . . 11
13 eleq1 2220 . . . . . . . . . . 11
1411, 12, 133orbi123d 1293 . . . . . . . . . 10
1510, 14imbi12d 233 . . . . . . . . 9
16 ordtriexmid.1 . . . . . . . . . 10
1716rspec2 2546 . . . . . . . . 9
188, 15, 17vtocl 2766 . . . . . . . 8
197, 18mpan2 422 . . . . . . 7
206, 19vtoclga 2778 . . . . . 6
212, 20ax-mp 5 . . . . 5
22 3orass 966 . . . . 5
2321, 22mpbi 144 . . . 4
241, 23mtpor 1407 . . 3
25 ordtriexmidlem2 4478 . . . 4
268snid 3591 . . . . . 6
27 biidd 171 . . . . . . 7
2827elrab3 2869 . . . . . 6
2926, 28ax-mp 5 . . . . 5
3029biimpi 119 . . . 4
3125, 30orim12i 749 . . 3
3224, 31ax-mp 5 . 2
33 orcom 718 . 2
3432, 33mpbir 145 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 103   wb 104   wo 698   w3o 962   wceq 1335   wcel 2128  wral 2435  crab 2439  c0 3394  csn 3560  con0 4323 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-14 2131  ax-ext 2139  ax-sep 4082  ax-nul 4090  ax-pow 4135 This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  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-nul 3395  df-pw 3545  df-sn 3566  df-uni 3773  df-tr 4063  df-iord 4326  df-on 4328  df-suc 4331 This theorem is referenced by: (None)
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