Theorem ordtriexmidlem2 4552

Description: Lemma for decidability and ordinals. The set {𝑥 ∈ {∅} ∣ 𝜑} is a way of connecting statements about ordinals (such as trichotomy in ordtriexmid 4553 or weak linearity in ordsoexmid 4594) 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

Step	Hyp	Ref	Expression
1		noel 3450	. . 3 ⊢ ¬ ∅ ∈ ∅
2		eleq2 2257	. . 3 ⊢ ({𝑥 ∈ {∅} ∣ 𝜑} = ∅ → (∅ ∈ {𝑥 ∈ {∅} ∣ 𝜑} ↔ ∅ ∈ ∅))
3	1, 2	mtbiri 676	. 2 ⊢ ({𝑥 ∈ {∅} ∣ 𝜑} = ∅ → ¬ ∅ ∈ {𝑥 ∈ {∅} ∣ 𝜑})
4		0ex 4156	. . . 4 ⊢ ∅ ∈ V
5	4	snid 3649	. . 3 ⊢ ∅ ∈ {∅}
6		biidd 172	. . . 4 ⊢ (𝑥 = ∅ → (𝜑 ↔ 𝜑))
7	6	elrab3 2917	. . 3 ⊢ (∅ ∈ {∅} → (∅ ∈ {𝑥 ∈ {∅} ∣ 𝜑} ↔ 𝜑))
8	5, 7	ax-mp 5	. 2 ⊢ (∅ ∈ {𝑥 ∈ {∅} ∣ 𝜑} ↔ 𝜑)
9	3, 8	sylnib 677	1 ⊢ ({𝑥 ∈ {∅} ∣ 𝜑} = ∅ → ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 105   = wceq 1364   ∈ wcel 2164  {crab 2476  ∅c0 3446  {csn 3618

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 2175  ax-nul 4155

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rab 2481  df-v 2762  df-dif 3155  df-nul 3447  df-sn 3624

This theorem is referenced by:  ordtriexmid  4553  ontriexmidim  4554  ordtri2orexmid  4555  ontr2exmid  4557  onsucsssucexmid  4559  ordsoexmid  4594  0elsucexmid  4597  ordpwsucexmid  4602