Theorem ordtriexmidlem2 4436

Description: Lemma for decidability and ordinals. The set {𝑥 ∈ {∅} ∣ 𝜑} is a way of connecting statements about ordinals (such as trichotomy in ordtriexmid 4437 or weak linearity in ordsoexmid 4477) 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 3367	. . 3 ⊢ ¬ ∅ ∈ ∅
2		eleq2 2203	. . 3 ⊢ ({𝑥 ∈ {∅} ∣ 𝜑} = ∅ → (∅ ∈ {𝑥 ∈ {∅} ∣ 𝜑} ↔ ∅ ∈ ∅))
3	1, 2	mtbiri 664	. 2 ⊢ ({𝑥 ∈ {∅} ∣ 𝜑} = ∅ → ¬ ∅ ∈ {𝑥 ∈ {∅} ∣ 𝜑})
4		0ex 4055	. . . 4 ⊢ ∅ ∈ V
5	4	snid 3556	. . 3 ⊢ ∅ ∈ {∅}
6		biidd 171	. . . 4 ⊢ (𝑥 = ∅ → (𝜑 ↔ 𝜑))
7	6	elrab3 2841	. . 3 ⊢ (∅ ∈ {∅} → (∅ ∈ {𝑥 ∈ {∅} ∣ 𝜑} ↔ 𝜑))
8	5, 7	ax-mp 5	. 2 ⊢ (∅ ∈ {𝑥 ∈ {∅} ∣ 𝜑} ↔ 𝜑)
9	3, 8	sylnib 665	1 ⊢ ({𝑥 ∈ {∅} ∣ 𝜑} = ∅ → ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 104   = wceq 1331   ∈ wcel 1480  {crab 2420  ∅c0 3363  {csn 3527

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-nul 4054

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rab 2425  df-v 2688  df-dif 3073  df-nul 3364  df-sn 3533

This theorem is referenced by:  ordtriexmid  4437  ordtri2orexmid  4438  ontr2exmid  4440  onsucsssucexmid  4442  ordsoexmid  4477  0elsucexmid  4480  ordpwsucexmid  4485