Theorem ordtriexmidlem 4336

Description: Lemma for decidability and ordinals. The set {𝑥 ∈ {∅} ∣ 𝜑} is a way of connecting statements about ordinals (such as trichotomy in ordtriexmid 4338 or weak linearity in ordsoexmid 4378) with a proposition 𝜑. Our lemma states that it is an ordinal number. (Contributed by Jim Kingdon, 28-Jan-2019.)

Assertion

Ref	Expression
ordtriexmidlem	⊢ {𝑥 ∈ {∅} ∣ 𝜑} ∈ On

Proof of Theorem ordtriexmidlem

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 107	. . . . . 6 ⊢ ((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ {𝑥 ∈ {∅} ∣ 𝜑}) → 𝑦 ∈ 𝑧)
2		elrabi 2768	. . . . . . . . 9 ⊢ (𝑧 ∈ {𝑥 ∈ {∅} ∣ 𝜑} → 𝑧 ∈ {∅})
3		velsn 3463	. . . . . . . . 9 ⊢ (𝑧 ∈ {∅} ↔ 𝑧 = ∅)
4	2, 3	sylib 120	. . . . . . . 8 ⊢ (𝑧 ∈ {𝑥 ∈ {∅} ∣ 𝜑} → 𝑧 = ∅)
5		noel 3290	. . . . . . . . 9 ⊢ ¬ 𝑦 ∈ ∅
6		eleq2 2151	. . . . . . . . 9 ⊢ (𝑧 = ∅ → (𝑦 ∈ 𝑧 ↔ 𝑦 ∈ ∅))
7	5, 6	mtbiri 635	. . . . . . . 8 ⊢ (𝑧 = ∅ → ¬ 𝑦 ∈ 𝑧)
8	4, 7	syl 14	. . . . . . 7 ⊢ (𝑧 ∈ {𝑥 ∈ {∅} ∣ 𝜑} → ¬ 𝑦 ∈ 𝑧)
9	8	adantl 271	. . . . . 6 ⊢ ((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ {𝑥 ∈ {∅} ∣ 𝜑}) → ¬ 𝑦 ∈ 𝑧)
10	1, 9	pm2.21dd 585	. . . . 5 ⊢ ((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ {𝑥 ∈ {∅} ∣ 𝜑}) → 𝑦 ∈ {𝑥 ∈ {∅} ∣ 𝜑})
11	10	gen2 1384	. . . 4 ⊢ ∀𝑦∀𝑧((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ {𝑥 ∈ {∅} ∣ 𝜑}) → 𝑦 ∈ {𝑥 ∈ {∅} ∣ 𝜑})
12		dftr2 3938	. . . 4 ⊢ (Tr {𝑥 ∈ {∅} ∣ 𝜑} ↔ ∀𝑦∀𝑧((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ {𝑥 ∈ {∅} ∣ 𝜑}) → 𝑦 ∈ {𝑥 ∈ {∅} ∣ 𝜑}))
13	11, 12	mpbir 144	. . 3 ⊢ Tr {𝑥 ∈ {∅} ∣ 𝜑}
14		ssrab2 3106	. . 3 ⊢ {𝑥 ∈ {∅} ∣ 𝜑} ⊆ {∅}
15		ord0 4218	. . . . 5 ⊢ Ord ∅
16		ordsucim 4317	. . . . 5 ⊢ (Ord ∅ → Ord suc ∅)
17	15, 16	ax-mp 7	. . . 4 ⊢ Ord suc ∅
18		suc0 4238	. . . . 5 ⊢ suc ∅ = {∅}
19		ordeq 4199	. . . . 5 ⊢ (suc ∅ = {∅} → (Ord suc ∅ ↔ Ord {∅}))
20	18, 19	ax-mp 7	. . . 4 ⊢ (Ord suc ∅ ↔ Ord {∅})
21	17, 20	mpbi 143	. . 3 ⊢ Ord {∅}
22		trssord 4207	. . 3 ⊢ ((Tr {𝑥 ∈ {∅} ∣ 𝜑} ∧ {𝑥 ∈ {∅} ∣ 𝜑} ⊆ {∅} ∧ Ord {∅}) → Ord {𝑥 ∈ {∅} ∣ 𝜑})
23	13, 14, 21, 22	mp3an 1273	. 2 ⊢ Ord {𝑥 ∈ {∅} ∣ 𝜑}
24		p0ex 4023	. . . 4 ⊢ {∅} ∈ V
25	24	rabex 3983	. . 3 ⊢ {𝑥 ∈ {∅} ∣ 𝜑} ∈ V
26	25	elon 4201	. 2 ⊢ ({𝑥 ∈ {∅} ∣ 𝜑} ∈ On ↔ Ord {𝑥 ∈ {∅} ∣ 𝜑})
27	23, 26	mpbir 144	1 ⊢ {𝑥 ∈ {∅} ∣ 𝜑} ∈ On

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 102   ↔ wb 103  ∀wal 1287   = wceq 1289   ∈ wcel 1438  {crab 2363   ⊆ wss 2999  ∅c0 3286  {csn 3446  Tr wtr 3936  Ord word 4189  Oncon0 4190  suc csuc 4192

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-nul 3965  ax-pow 4009

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-uni 3654  df-tr 3937  df-iord 4193  df-on 4195  df-suc 4198

This theorem is referenced by:  ordtriexmid  4338  ordtri2orexmid  4339  ontr2exmid  4341  onsucsssucexmid  4343  ordsoexmid  4378  0elsucexmid  4381  ordpwsucexmid  4386  unfiexmid  6626