Theorem ordtriexmidlem2 4504

Description: Lemma for decidability and ordinals. The set {𝑥 ∈ {∅} ∣ 𝜑} is a way of connecting statements about ordinals (such as trichotomy in ordtriexmid 4505 or weak linearity in ordsoexmid 4546) 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 3418	. . 3 ⊢ ¬ ∅ ∈ ∅
2		eleq2 2234	. . 3 ⊢ ({𝑥 ∈ {∅} ∣ 𝜑} = ∅ → (∅ ∈ {𝑥 ∈ {∅} ∣ 𝜑} ↔ ∅ ∈ ∅))
3	1, 2	mtbiri 670	. 2 ⊢ ({𝑥 ∈ {∅} ∣ 𝜑} = ∅ → ¬ ∅ ∈ {𝑥 ∈ {∅} ∣ 𝜑})
4		0ex 4116	. . . 4 ⊢ ∅ ∈ V
5	4	snid 3614	. . 3 ⊢ ∅ ∈ {∅}
6		biidd 171	. . . 4 ⊢ (𝑥 = ∅ → (𝜑 ↔ 𝜑))
7	6	elrab3 2887	. . 3 ⊢ (∅ ∈ {∅} → (∅ ∈ {𝑥 ∈ {∅} ∣ 𝜑} ↔ 𝜑))
8	5, 7	ax-mp 5	. 2 ⊢ (∅ ∈ {𝑥 ∈ {∅} ∣ 𝜑} ↔ 𝜑)
9	3, 8	sylnib 671	1 ⊢ ({𝑥 ∈ {∅} ∣ 𝜑} = ∅ → ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 104   = wceq 1348   ∈ wcel 2141  {crab 2452  ∅c0 3414  {csn 3583

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152  ax-nul 4115

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rab 2457  df-v 2732  df-dif 3123  df-nul 3415  df-sn 3589

This theorem is referenced by:  ordtriexmid  4505  ontriexmidim  4506  ordtri2orexmid  4507  ontr2exmid  4509  onsucsssucexmid  4511  ordsoexmid  4546  0elsucexmid  4549  ordpwsucexmid  4554