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Theorem ordtriexmidlem2 4274
Description: Lemma for decidability and ordinals. The set {𝑥 ∈ {∅} ∣ 𝜑} is a way of connecting statements about ordinals (such as trichotomy in ordtriexmid 4275 or weak linearity in ordsoexmid 4314) 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 3256 . . 3 ¬ ∅ ∈ ∅
2 eleq2 2117 . . 3 ({𝑥 ∈ {∅} ∣ 𝜑} = ∅ → (∅ ∈ {𝑥 ∈ {∅} ∣ 𝜑} ↔ ∅ ∈ ∅))
31, 2mtbiri 610 . 2 ({𝑥 ∈ {∅} ∣ 𝜑} = ∅ → ¬ ∅ ∈ {𝑥 ∈ {∅} ∣ 𝜑})
4 0ex 3912 . . . 4 ∅ ∈ V
54snid 3430 . . 3 ∅ ∈ {∅}
6 biidd 165 . . . 4 (𝑥 = ∅ → (𝜑𝜑))
76elrab3 2722 . . 3 (∅ ∈ {∅} → (∅ ∈ {𝑥 ∈ {∅} ∣ 𝜑} ↔ 𝜑))
85, 7ax-mp 7 . 2 (∅ ∈ {𝑥 ∈ {∅} ∣ 𝜑} ↔ 𝜑)
93, 8sylnib 611 1 ({𝑥 ∈ {∅} ∣ 𝜑} = ∅ → ¬ 𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 102   = wceq 1259  wcel 1409  {crab 2327  c0 3252  {csn 3403
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-nul 3911
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rab 2332  df-v 2576  df-dif 2948  df-nul 3253  df-sn 3409
This theorem is referenced by:  ordtriexmid  4275  ordtri2orexmid  4276  ontr2exmid  4278  onsucsssucexmid  4280  ordsoexmid  4314  0elsucexmid  4317  ordpwsucexmid  4322
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