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Theorem ordtriexmidlem2 4209
Description: Lemma for decidability and ordinals. The set {x {∅} ∣ φ} is a way of connecting statements about ordinals (such as trichotomy in ordtriexmid 4210 or weak linearity in ordsoexmid 4240) with a proposition φ. Our lemma helps connect that set to excluded middle. (Contributed by Jim Kingdon, 28-Jan-2019.)
Assertion
Ref Expression
ordtriexmidlem2 ({x {∅} ∣ φ} = ∅ → ¬ φ)
Distinct variable group:   φ,x

Proof of Theorem ordtriexmidlem2
StepHypRef Expression
1 noel 3222 . . 3 ¬ ∅
2 eleq2 2098 . . 3 ({x {∅} ∣ φ} = ∅ → (∅ {x {∅} ∣ φ} ↔ ∅ ∅))
31, 2mtbiri 599 . 2 ({x {∅} ∣ φ} = ∅ → ¬ ∅ {x {∅} ∣ φ})
4 0ex 3875 . . . 4 V
54snid 3394 . . 3 {∅}
6 biidd 161 . . . 4 (x = ∅ → (φφ))
76elrab3 2693 . . 3 (∅ {∅} → (∅ {x {∅} ∣ φ} ↔ φ))
85, 7ax-mp 7 . 2 (∅ {x {∅} ∣ φ} ↔ φ)
93, 8sylnib 600 1 ({x {∅} ∣ φ} = ∅ → ¬ φ)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 98   = wceq 1242   wcel 1390  {crab 2304  c0 3218  {csn 3367
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-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-nul 3874
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rab 2309  df-v 2553  df-dif 2914  df-nul 3219  df-sn 3373
This theorem is referenced by:  ordtriexmid  4210  ordtri2orexmid  4211  onsucsssucexmid  4212  ordsoexmid  4240  ordpwsucexmid  4246
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