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Theorem ordtriexmidlem2 4502
Description: Lemma for decidability and ordinals. The set {𝑥 ∈ {∅} ∣ 𝜑} is a way of connecting statements about ordinals (such as trichotomy in ordtriexmid 4503 or weak linearity in ordsoexmid 4544) 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 3418 . . 3 ¬ ∅ ∈ ∅
2 eleq2 2234 . . 3 ({𝑥 ∈ {∅} ∣ 𝜑} = ∅ → (∅ ∈ {𝑥 ∈ {∅} ∣ 𝜑} ↔ ∅ ∈ ∅))
31, 2mtbiri 670 . 2 ({𝑥 ∈ {∅} ∣ 𝜑} = ∅ → ¬ ∅ ∈ {𝑥 ∈ {∅} ∣ 𝜑})
4 0ex 4114 . . . 4 ∅ ∈ V
54snid 3612 . . 3 ∅ ∈ {∅}
6 biidd 171 . . . 4 (𝑥 = ∅ → (𝜑𝜑))
76elrab3 2887 . . 3 (∅ ∈ {∅} → (∅ ∈ {𝑥 ∈ {∅} ∣ 𝜑} ↔ 𝜑))
85, 7ax-mp 5 . 2 (∅ ∈ {𝑥 ∈ {∅} ∣ 𝜑} ↔ 𝜑)
93, 8sylnib 671 1 ({𝑥 ∈ {∅} ∣ 𝜑} = ∅ → ¬ 𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 104   = wceq 1348  wcel 2141  {crab 2452  c0 3414  {csn 3581
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152  ax-nul 4113
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rab 2457  df-v 2732  df-dif 3123  df-nul 3415  df-sn 3587
This theorem is referenced by:  ordtriexmid  4503  ontriexmidim  4504  ordtri2orexmid  4505  ontr2exmid  4507  onsucsssucexmid  4509  ordsoexmid  4544  0elsucexmid  4547  ordpwsucexmid  4552
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