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Theorem ordtriexmidlem2 4537
Description: Lemma for decidability and ordinals. The set {𝑥 ∈ {∅} ∣ 𝜑} is a way of connecting statements about ordinals (such as trichotomy in ordtriexmid 4538 or weak linearity in ordsoexmid 4579) 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 3441 . . 3 ¬ ∅ ∈ ∅
2 eleq2 2253 . . 3 ({𝑥 ∈ {∅} ∣ 𝜑} = ∅ → (∅ ∈ {𝑥 ∈ {∅} ∣ 𝜑} ↔ ∅ ∈ ∅))
31, 2mtbiri 676 . 2 ({𝑥 ∈ {∅} ∣ 𝜑} = ∅ → ¬ ∅ ∈ {𝑥 ∈ {∅} ∣ 𝜑})
4 0ex 4145 . . . 4 ∅ ∈ V
54snid 3638 . . 3 ∅ ∈ {∅}
6 biidd 172 . . . 4 (𝑥 = ∅ → (𝜑𝜑))
76elrab3 2909 . . 3 (∅ ∈ {∅} → (∅ ∈ {𝑥 ∈ {∅} ∣ 𝜑} ↔ 𝜑))
85, 7ax-mp 5 . 2 (∅ ∈ {𝑥 ∈ {∅} ∣ 𝜑} ↔ 𝜑)
93, 8sylnib 677 1 ({𝑥 ∈ {∅} ∣ 𝜑} = ∅ → ¬ 𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 105   = wceq 1364  wcel 2160  {crab 2472  c0 3437  {csn 3607
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171  ax-nul 4144
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rab 2477  df-v 2754  df-dif 3146  df-nul 3438  df-sn 3613
This theorem is referenced by:  ordtriexmid  4538  ontriexmidim  4539  ordtri2orexmid  4540  ontr2exmid  4542  onsucsssucexmid  4544  ordsoexmid  4579  0elsucexmid  4582  ordpwsucexmid  4587
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