Theorem ordtriexmidlem 4556

Description: Lemma for decidability and ordinals. The set {𝑥 ∈ {∅} ∣ 𝜑} is a way of connecting statements about ordinals (such as trichotomy in ordtriexmid 4558 or weak linearity in ordsoexmid 4599) 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 2917	. . . . . . . . 9 ⊢ (𝑧 ∈ {𝑥 ∈ {∅} ∣ 𝜑} → 𝑧 ∈ {∅})
3		velsn 3640	. . . . . . . . 9 ⊢ (𝑧 ∈ {∅} ↔ 𝑧 = ∅)
4	2, 3	sylib 122	. . . . . . . 8 ⊢ (𝑧 ∈ {𝑥 ∈ {∅} ∣ 𝜑} → 𝑧 = ∅)
5		noel 3455	. . . . . . . . 9 ⊢ ¬ 𝑦 ∈ ∅
6		eleq2 2260	. . . . . . . . 9 ⊢ (𝑧 = ∅ → (𝑦 ∈ 𝑧 ↔ 𝑦 ∈ ∅))
7	5, 6	mtbiri 676	. . . . . . . 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 621	. . . . 5 ⊢ ((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ {𝑥 ∈ {∅} ∣ 𝜑}) → 𝑦 ∈ {𝑥 ∈ {∅} ∣ 𝜑})
11	10	gen2 1464	. . . 4 ⊢ ∀𝑦∀𝑧((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ {𝑥 ∈ {∅} ∣ 𝜑}) → 𝑦 ∈ {𝑥 ∈ {∅} ∣ 𝜑})
12		dftr2 4134	. . . 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 3269	. . 3 ⊢ {𝑥 ∈ {∅} ∣ 𝜑} ⊆ {∅}
15		ord0 4427	. . . . 5 ⊢ Ord ∅
16		ordsucim 4537	. . . . 5 ⊢ (Ord ∅ → Ord suc ∅)
17	15, 16	ax-mp 5	. . . 4 ⊢ Ord suc ∅
18		suc0 4447	. . . . 5 ⊢ suc ∅ = {∅}
19		ordeq 4408	. . . . 5 ⊢ (suc ∅ = {∅} → (Ord suc ∅ ↔ Ord {∅}))
20	18, 19	ax-mp 5	. . . 4 ⊢ (Ord suc ∅ ↔ Ord {∅})
21	17, 20	mpbi 145	. . 3 ⊢ Ord {∅}
22		trssord 4416	. . 3 ⊢ ((Tr {𝑥 ∈ {∅} ∣ 𝜑} ∧ {𝑥 ∈ {∅} ∣ 𝜑} ⊆ {∅} ∧ Ord {∅}) → Ord {𝑥 ∈ {∅} ∣ 𝜑})
23	13, 14, 21, 22	mp3an 1348	. 2 ⊢ Ord {𝑥 ∈ {∅} ∣ 𝜑}
24		p0ex 4222	. . . 4 ⊢ {∅} ∈ V
25	24	rabex 4178	. . 3 ⊢ {𝑥 ∈ {∅} ∣ 𝜑} ∈ V
26	25	elon 4410	. 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 1362   = wceq 1364   ∈ wcel 2167  {crab 2479   ⊆ wss 3157  ∅c0 3451  {csn 3623  Tr wtr 4132  Ord word 4398  Oncon0 4399  suc csuc 4401

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-nul 4160  ax-pow 4208

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-uni 3841  df-tr 4133  df-iord 4402  df-on 4404  df-suc 4407

This theorem is referenced by:  ordtriexmid  4558  ontriexmidim  4559  ordtri2orexmid  4560  ontr2exmid  4562  onsucsssucexmid  4564  ordsoexmid  4599  0elsucexmid  4602  ordpwsucexmid  4607  unfiexmid  6988  exmidonfinlem  7272