Theorem ordtriexmidlem 4640

Description: Lemma for decidability and ordinals. The set {𝑥 ∈ {∅} ∣ 𝜑} is a way of connecting statements about ordinals (such as trichotomy in ordtriexmid 4642 or weak linearity in ordsoexmid 4683) 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 109	. . . . . 6 ⊢ ((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ {𝑥 ∈ {∅} ∣ 𝜑}) → 𝑦 ∈ 𝑧)
2		elrabi 2969	. . . . . . . . 9 ⊢ (𝑧 ∈ {𝑥 ∈ {∅} ∣ 𝜑} → 𝑧 ∈ {∅})
3		velsn 3705	. . . . . . . . 9 ⊢ (𝑧 ∈ {∅} ↔ 𝑧 = ∅)
4	2, 3	sylib 122	. . . . . . . 8 ⊢ (𝑧 ∈ {𝑥 ∈ {∅} ∣ 𝜑} → 𝑧 = ∅)
5		noel 3511	. . . . . . . . 9 ⊢ ¬ 𝑦 ∈ ∅
6		eleq2 2296	. . . . . . . . 9 ⊢ (𝑧 = ∅ → (𝑦 ∈ 𝑧 ↔ 𝑦 ∈ ∅))
7	5, 6	mtbiri 682	. . . . . . . 8 ⊢ (𝑧 = ∅ → ¬ 𝑦 ∈ 𝑧)
8	4, 7	syl 14	. . . . . . 7 ⊢ (𝑧 ∈ {𝑥 ∈ {∅} ∣ 𝜑} → ¬ 𝑦 ∈ 𝑧)
9	8	adantl 277	. . . . . 6 ⊢ ((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ {𝑥 ∈ {∅} ∣ 𝜑}) → ¬ 𝑦 ∈ 𝑧)
10	1, 9	pm2.21dd 625	. . . . 5 ⊢ ((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ {𝑥 ∈ {∅} ∣ 𝜑}) → 𝑦 ∈ {𝑥 ∈ {∅} ∣ 𝜑})
11	10	gen2 1499	. . . 4 ⊢ ∀𝑦∀𝑧((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ {𝑥 ∈ {∅} ∣ 𝜑}) → 𝑦 ∈ {𝑥 ∈ {∅} ∣ 𝜑})
12		dftr2 4209	. . . 4 ⊢ (Tr {𝑥 ∈ {∅} ∣ 𝜑} ↔ ∀𝑦∀𝑧((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ {𝑥 ∈ {∅} ∣ 𝜑}) → 𝑦 ∈ {𝑥 ∈ {∅} ∣ 𝜑}))
13	11, 12	mpbir 146	. . 3 ⊢ Tr {𝑥 ∈ {∅} ∣ 𝜑}
14		ssrab2 3322	. . 3 ⊢ {𝑥 ∈ {∅} ∣ 𝜑} ⊆ {∅}
15		ord0 4511	. . . . 5 ⊢ Ord ∅
16		ordsucim 4621	. . . . 5 ⊢ (Ord ∅ → Ord suc ∅)
17	15, 16	ax-mp 5	. . . 4 ⊢ Ord suc ∅
18		suc0 4531	. . . . 5 ⊢ suc ∅ = {∅}
19		ordeq 4492	. . . . 5 ⊢ (suc ∅ = {∅} → (Ord suc ∅ ↔ Ord {∅}))
20	18, 19	ax-mp 5	. . . 4 ⊢ (Ord suc ∅ ↔ Ord {∅})
21	17, 20	mpbi 145	. . 3 ⊢ Ord {∅}
22		trssord 4500	. . 3 ⊢ ((Tr {𝑥 ∈ {∅} ∣ 𝜑} ∧ {𝑥 ∈ {∅} ∣ 𝜑} ⊆ {∅} ∧ Ord {∅}) → Ord {𝑥 ∈ {∅} ∣ 𝜑})
23	13, 14, 21, 22	mp3an 1374	. 2 ⊢ Ord {𝑥 ∈ {∅} ∣ 𝜑}
24		p0ex 4300	. . . 4 ⊢ {∅} ∈ V
25	24	rabex 4255	. . 3 ⊢ {𝑥 ∈ {∅} ∣ 𝜑} ∈ V
26	25	elon 4494	. 2 ⊢ ({𝑥 ∈ {∅} ∣ 𝜑} ∈ On ↔ Ord {𝑥 ∈ {∅} ∣ 𝜑})
27	23, 26	mpbir 146	1 ⊢ {𝑥 ∈ {∅} ∣ 𝜑} ∈ On

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1396   = wceq 1398   ∈ wcel 2203  {crab 2524   ⊆ wss 3210  ∅c0 3507  {csn 3688  Tr wtr 4207  Ord word 4482  Oncon0 4483  suc csuc 4485

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-14 2206  ax-ext 2214  ax-sep 4227  ax-nul 4235  ax-pow 4286

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2814  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-pw 3670  df-sn 3694  df-uni 3914  df-tr 4208  df-iord 4486  df-on 4488  df-suc 4491

This theorem is referenced by:  ordtriexmid  4642  ontriexmidim  4643  ordtri2orexmid  4644  ontr2exmid  4646  onsucsssucexmid  4648  ordsoexmid  4683  0elsucexmid  4686  ordpwsucexmid  4691  unfiexmid  7177  exmidonfinlem  7495