Theorem ordtriexmidlem2 4624

Description: Lemma for decidability and ordinals. The set {𝑥 ∈ {∅} ∣ 𝜑} is a way of connecting statements about ordinals (such as trichotomy in ordtriexmid 4625 or weak linearity in ordsoexmid 4666) 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 3500	. . 3 ⊢ ¬ ∅ ∈ ∅
2		eleq2 2295	. . 3 ⊢ ({𝑥 ∈ {∅} ∣ 𝜑} = ∅ → (∅ ∈ {𝑥 ∈ {∅} ∣ 𝜑} ↔ ∅ ∈ ∅))
3	1, 2	mtbiri 682	. 2 ⊢ ({𝑥 ∈ {∅} ∣ 𝜑} = ∅ → ¬ ∅ ∈ {𝑥 ∈ {∅} ∣ 𝜑})
4		0ex 4221	. . . 4 ⊢ ∅ ∈ V
5	4	snid 3704	. . 3 ⊢ ∅ ∈ {∅}
6		biidd 172	. . . 4 ⊢ (𝑥 = ∅ → (𝜑 ↔ 𝜑))
7	6	elrab3 2964	. . 3 ⊢ (∅ ∈ {∅} → (∅ ∈ {𝑥 ∈ {∅} ∣ 𝜑} ↔ 𝜑))
8	5, 7	ax-mp 5	. 2 ⊢ (∅ ∈ {𝑥 ∈ {∅} ∣ 𝜑} ↔ 𝜑)
9	3, 8	sylnib 683	1 ⊢ ({𝑥 ∈ {∅} ∣ 𝜑} = ∅ → ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 105   = wceq 1398   ∈ wcel 2202  {crab 2515  ∅c0 3496  {csn 3673

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213  ax-nul 4220

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-rab 2520  df-v 2805  df-dif 3203  df-nul 3497  df-sn 3679

This theorem is referenced by:  ordtriexmid  4625  ontriexmidim  4626  ordtri2orexmid  4627  ontr2exmid  4629  onsucsssucexmid  4631  ordsoexmid  4666  0elsucexmid  4669  ordpwsucexmid  4674