Theorem ordtriexmidlem 4664

Description: Lemma for decidability and ordinals. The set {𝑥 ∈ {∅} ∣ 𝜑} is a way of connecting statements about ordinals (such as trichotomy in ordtriexmid 4666 or weak linearity in ordsoexmid 4707) 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 2979	. . . . . . . . 9 ⊢ (𝑧 ∈ {𝑥 ∈ {∅} ∣ 𝜑} → 𝑧 ∈ {∅})
3		velsn 3725	. . . . . . . . 9 ⊢ (𝑧 ∈ {∅} ↔ 𝑧 = ∅)
4	2, 3	sylib 122	. . . . . . . 8 ⊢ (𝑧 ∈ {𝑥 ∈ {∅} ∣ 𝜑} → 𝑧 = ∅)
5		noel 3525	. . . . . . . . 9 ⊢ ¬ 𝑦 ∈ ∅
6		eleq2 2302	. . . . . . . . 9 ⊢ (𝑧 = ∅ → (𝑦 ∈ 𝑧 ↔ 𝑦 ∈ ∅))
7	5, 6	mtbiri 686	. . . . . . . 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 629	. . . . 5 ⊢ ((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ {𝑥 ∈ {∅} ∣ 𝜑}) → 𝑦 ∈ {𝑥 ∈ {∅} ∣ 𝜑})
11	10	gen2 1503	. . . 4 ⊢ ∀𝑦∀𝑧((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ {𝑥 ∈ {∅} ∣ 𝜑}) → 𝑦 ∈ {𝑥 ∈ {∅} ∣ 𝜑})
12		dftr2 4229	. . . 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 3333	. . 3 ⊢ {𝑥 ∈ {∅} ∣ 𝜑} ⊆ {∅}
15		ord0 4534	. . . . 5 ⊢ Ord ∅
16		ordsucim 4645	. . . . 5 ⊢ (Ord ∅ → Ord suc ∅)
17	15, 16	ax-mp 5	. . . 4 ⊢ Ord suc ∅
18		suc0 4554	. . . . 5 ⊢ suc ∅ = {∅}
19		ordeq 4515	. . . . 5 ⊢ (suc ∅ = {∅} → (Ord suc ∅ ↔ Ord {∅}))
20	18, 19	ax-mp 5	. . . 4 ⊢ (Ord suc ∅ ↔ Ord {∅})
21	17, 20	mpbi 145	. . 3 ⊢ Ord {∅}
22		trssord 4523	. . 3 ⊢ ((Tr {𝑥 ∈ {∅} ∣ 𝜑} ∧ {𝑥 ∈ {∅} ∣ 𝜑} ⊆ {∅} ∧ Ord {∅}) → Ord {𝑥 ∈ {∅} ∣ 𝜑})
23	13, 14, 21, 22	mp3an 1378	. 2 ⊢ Ord {𝑥 ∈ {∅} ∣ 𝜑}
24		p0ex 4323	. . . 4 ⊢ {∅} ∈ V
25	24	rabex 4278	. . 3 ⊢ {𝑥 ∈ {∅} ∣ 𝜑} ∈ V
26	25	elon 4517	. 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 1400   = wceq 1402   ∈ wcel 2209  {crab 2532   ⊆ wss 3220  ∅c0 3520  {csn 3708  Tr wtr 4227  Ord word 4505  Oncon0 4506  suc csuc 4508

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-nul 4257  ax-pow 4309

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3714  df-uni 3934  df-tr 4228  df-iord 4509  df-on 4511  df-suc 4514

This theorem is referenced by:  ordtriexmid  4666  ontriexmidim  4667  ordtri2orexmid  4668  ontr2exmid  4670  onsucsssucexmid  4672  ordsoexmid  4707  0elsucexmid  4710  ordpwsucexmid  4715  unfiexmid  7219  exmidonfinlem  7539