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Theorem ordtriexmidlem2 4374
 Description: Lemma for decidability and ordinals. The set {𝑥 ∈ {∅} ∣ 𝜑} is a way of connecting statements about ordinals (such as trichotomy in ordtriexmid 4375 or weak linearity in ordsoexmid 4415) 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 3314 . . 3 ¬ ∅ ∈ ∅
2 eleq2 2163 . . 3 ({𝑥 ∈ {∅} ∣ 𝜑} = ∅ → (∅ ∈ {𝑥 ∈ {∅} ∣ 𝜑} ↔ ∅ ∈ ∅))
31, 2mtbiri 641 . 2 ({𝑥 ∈ {∅} ∣ 𝜑} = ∅ → ¬ ∅ ∈ {𝑥 ∈ {∅} ∣ 𝜑})
4 0ex 3995 . . . 4 ∅ ∈ V
54snid 3503 . . 3 ∅ ∈ {∅}
6 biidd 171 . . . 4 (𝑥 = ∅ → (𝜑𝜑))
76elrab3 2794 . . 3 (∅ ∈ {∅} → (∅ ∈ {𝑥 ∈ {∅} ∣ 𝜑} ↔ 𝜑))
85, 7ax-mp 7 . 2 (∅ ∈ {𝑥 ∈ {∅} ∣ 𝜑} ↔ 𝜑)
93, 8sylnib 642 1 ({𝑥 ∈ {∅} ∣ 𝜑} = ∅ → ¬ 𝜑)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 104   = wceq 1299   ∈ wcel 1448  {crab 2379  ∅c0 3310  {csn 3474 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-nul 3994 This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-rab 2384  df-v 2643  df-dif 3023  df-nul 3311  df-sn 3480 This theorem is referenced by:  ordtriexmid  4375  ordtri2orexmid  4376  ontr2exmid  4378  onsucsssucexmid  4380  ordsoexmid  4415  0elsucexmid  4418  ordpwsucexmid  4423
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