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Theorem ordtriexmidlem2 4193
Description: Lemma for decidability and ordinals. The set {x {∅} ∣ φ} is a way of connecting statements about ordinals (such as trichotomy in ordtriexmid 4194 or weak linearity in ordsoexmid 4224) 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 3205 . . 3 ¬ ∅
2 eleq2 2083 . . 3 ({x {∅} ∣ φ} = ∅ → (∅ {x {∅} ∣ φ} ↔ ∅ ∅))
31, 2mtbiri 587 . 2 ({x {∅} ∣ φ} = ∅ → ¬ ∅ {x {∅} ∣ φ})
4 0ex 3858 . . . 4 V
54snid 3377 . . 3 {∅}
6 biidd 161 . . . 4 (x = ∅ → (φφ))
76elrab3 2676 . . 3 (∅ {∅} → (∅ {x {∅} ∣ φ} ↔ φ))
85, 7ax-mp 7 . 2 (∅ {x {∅} ∣ φ} ↔ φ)
93, 8sylnib 588 1 ({x {∅} ∣ φ} = ∅ → ¬ φ)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 98   = wceq 1228   wcel 1374  {crab 2288  c0 3201  {csn 3350
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-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-nul 3857
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-rab 2293  df-v 2537  df-dif 2897  df-nul 3202  df-sn 3356
This theorem is referenced by:  ordtriexmid  4194  ordtri2orexmid  4195  onsucsssucexmid  4196  ordsoexmid  4224  ordpwsucexmid  4230
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