Theorem ordtriexmidlem 4530

Description: Lemma for decidability and ordinals. The set {𝑥 ∈ {∅} ∣ 𝜑} is a way of connecting statements about ordinals (such as trichotomy in ordtriexmid 4532 or weak linearity in ordsoexmid 4573) 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 2902	. . . . . . . . 9 ⊢ (𝑧 ∈ {𝑥 ∈ {∅} ∣ 𝜑} → 𝑧 ∈ {∅})
3		velsn 3621	. . . . . . . . 9 ⊢ (𝑧 ∈ {∅} ↔ 𝑧 = ∅)
4	2, 3	sylib 122	. . . . . . . 8 ⊢ (𝑧 ∈ {𝑥 ∈ {∅} ∣ 𝜑} → 𝑧 = ∅)
5		noel 3438	. . . . . . . . 9 ⊢ ¬ 𝑦 ∈ ∅
6		eleq2 2251	. . . . . . . . 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 1460	. . . 4 ⊢ ∀𝑦∀𝑧((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ {𝑥 ∈ {∅} ∣ 𝜑}) → 𝑦 ∈ {𝑥 ∈ {∅} ∣ 𝜑})
12		dftr2 4115	. . . 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 3252	. . 3 ⊢ {𝑥 ∈ {∅} ∣ 𝜑} ⊆ {∅}
15		ord0 4403	. . . . 5 ⊢ Ord ∅
16		ordsucim 4511	. . . . 5 ⊢ (Ord ∅ → Ord suc ∅)
17	15, 16	ax-mp 5	. . . 4 ⊢ Ord suc ∅
18		suc0 4423	. . . . 5 ⊢ suc ∅ = {∅}
19		ordeq 4384	. . . . 5 ⊢ (suc ∅ = {∅} → (Ord suc ∅ ↔ Ord {∅}))
20	18, 19	ax-mp 5	. . . 4 ⊢ (Ord suc ∅ ↔ Ord {∅})
21	17, 20	mpbi 145	. . 3 ⊢ Ord {∅}
22		trssord 4392	. . 3 ⊢ ((Tr {𝑥 ∈ {∅} ∣ 𝜑} ∧ {𝑥 ∈ {∅} ∣ 𝜑} ⊆ {∅} ∧ Ord {∅}) → Ord {𝑥 ∈ {∅} ∣ 𝜑})
23	13, 14, 21, 22	mp3an 1347	. 2 ⊢ Ord {𝑥 ∈ {∅} ∣ 𝜑}
24		p0ex 4200	. . . 4 ⊢ {∅} ∈ V
25	24	rabex 4159	. . 3 ⊢ {𝑥 ∈ {∅} ∣ 𝜑} ∈ V
26	25	elon 4386	. 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 1361   = wceq 1363   ∈ wcel 2158  {crab 2469   ⊆ wss 3141  ∅c0 3434  {csn 3604  Tr wtr 4113  Ord word 4374  Oncon0 4375  suc csuc 4377

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-nul 4141  ax-pow 4186

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-rab 2474  df-v 2751  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-uni 3822  df-tr 4114  df-iord 4378  df-on 4380  df-suc 4383

This theorem is referenced by:  ordtriexmid  4532  ontriexmidim  4533  ordtri2orexmid  4534  ontr2exmid  4536  onsucsssucexmid  4538  ordsoexmid  4573  0elsucexmid  4576  ordpwsucexmid  4581  unfiexmid  6930  exmidonfinlem  7205