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Theorem ordtriexmidlem2 4218
Description: Lemma for decidability and ordinals. The set {𝑥 ∈ {∅} ∣ 𝜑} is a way of connecting statements about ordinals (such as trichotomy in ordtriexmid 4219 or weak linearity in ordsoexmid 4256) 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 3225 . . 3 ¬ ∅ ∈ ∅
2 eleq2 2101 . . 3 ({𝑥 ∈ {∅} ∣ 𝜑} = ∅ → (∅ ∈ {𝑥 ∈ {∅} ∣ 𝜑} ↔ ∅ ∈ ∅))
31, 2mtbiri 600 . 2 ({𝑥 ∈ {∅} ∣ 𝜑} = ∅ → ¬ ∅ ∈ {𝑥 ∈ {∅} ∣ 𝜑})
4 0ex 3881 . . . 4 ∅ ∈ V
54snid 3399 . . 3 ∅ ∈ {∅}
6 biidd 161 . . . 4 (𝑥 = ∅ → (𝜑𝜑))
76elrab3 2696 . . 3 (∅ ∈ {∅} → (∅ ∈ {𝑥 ∈ {∅} ∣ 𝜑} ↔ 𝜑))
85, 7ax-mp 7 . 2 (∅ ∈ {𝑥 ∈ {∅} ∣ 𝜑} ↔ 𝜑)
93, 8sylnib 601 1 ({𝑥 ∈ {∅} ∣ 𝜑} = ∅ → ¬ 𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 98   = wceq 1243  wcel 1393  {crab 2307  c0 3221  {csn 3372
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 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-nul 3880
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rab 2312  df-v 2556  df-dif 2917  df-nul 3222  df-sn 3378
This theorem is referenced by:  ordtriexmid  4219  ordtri2orexmid  4220  ontr2exmid  4222  onsucsssucexmid  4224  ordsoexmid  4256  0elsucexmid  4259  ordpwsucexmid  4264
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