Theorem ordtriexmidlem2 4209

Description: Lemma for decidability and ordinals. The set {x ∈ {∅} ∣ φ} is a way of connecting statements about ordinals (such as trichotomy in ordtriexmid 4210 or weak linearity in ordsoexmid 4240) 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 3222	. . 3 ⊢ ¬ ∅ ∈ ∅
2		eleq2 2098	. . 3 ⊢ ({x ∈ {∅} ∣ φ} = ∅ → (∅ ∈ {x ∈ {∅} ∣ φ} ↔ ∅ ∈ ∅))
3	1, 2	mtbiri 599	. 2 ⊢ ({x ∈ {∅} ∣ φ} = ∅ → ¬ ∅ ∈ {x ∈ {∅} ∣ φ})
4		0ex 3875	. . . 4 ⊢ ∅ ∈ V
5	4	snid 3394	. . 3 ⊢ ∅ ∈ {∅}
6		biidd 161	. . . 4 ⊢ (x = ∅ → (φ ↔ φ))
7	6	elrab3 2693	. . 3 ⊢ (∅ ∈ {∅} → (∅ ∈ {x ∈ {∅} ∣ φ} ↔ φ))
8	5, 7	ax-mp 7	. 2 ⊢ (∅ ∈ {x ∈ {∅} ∣ φ} ↔ φ)
9	3, 8	sylnib 600	1 ⊢ ({x ∈ {∅} ∣ φ} = ∅ → ¬ φ)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 98   = wceq 1242   ∈ wcel 1390  {crab 2304  ∅c0 3218  {csn 3367

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-nul 3874

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rab 2309  df-v 2553  df-dif 2914  df-nul 3219  df-sn 3373

This theorem is referenced by:  ordtriexmid  4210  ordtri2orexmid  4211  onsucsssucexmid  4212  ordsoexmid  4240  ordpwsucexmid  4246