Theorem ordtriexmidlem2 4556

Description: Lemma for decidability and ordinals. The set {𝑥 ∈ {∅} ∣ 𝜑} is a way of connecting statements about ordinals (such as trichotomy in ordtriexmid 4557 or weak linearity in ordsoexmid 4598) 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 3454	. . 3 ⊢ ¬ ∅ ∈ ∅
2		eleq2 2260	. . 3 ⊢ ({𝑥 ∈ {∅} ∣ 𝜑} = ∅ → (∅ ∈ {𝑥 ∈ {∅} ∣ 𝜑} ↔ ∅ ∈ ∅))
3	1, 2	mtbiri 676	. 2 ⊢ ({𝑥 ∈ {∅} ∣ 𝜑} = ∅ → ¬ ∅ ∈ {𝑥 ∈ {∅} ∣ 𝜑})
4		0ex 4160	. . . 4 ⊢ ∅ ∈ V
5	4	snid 3653	. . 3 ⊢ ∅ ∈ {∅}
6		biidd 172	. . . 4 ⊢ (𝑥 = ∅ → (𝜑 ↔ 𝜑))
7	6	elrab3 2921	. . 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 2167  {crab 2479  ∅c0 3450  {csn 3622

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178  ax-nul 4159

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-rab 2484  df-v 2765  df-dif 3159  df-nul 3451  df-sn 3628

This theorem is referenced by:  ordtriexmid  4557  ontriexmidim  4558  ordtri2orexmid  4559  ontr2exmid  4561  onsucsssucexmid  4563  ordsoexmid  4598  0elsucexmid  4601  ordpwsucexmid  4606