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Theorem ordtriexmidlem 4551
Description: Lemma for decidability and ordinals. The set {𝑥 ∈ {∅} ∣ 𝜑} is a way of connecting statements about ordinals (such as trichotomy in ordtriexmid 4553 or weak linearity in ordsoexmid 4594) 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.
StepHypRef Expression
1 simpl 109 . . . . . 6 ((𝑦𝑧𝑧 ∈ {𝑥 ∈ {∅} ∣ 𝜑}) → 𝑦𝑧)
2 elrabi 2913 . . . . . . . . 9 (𝑧 ∈ {𝑥 ∈ {∅} ∣ 𝜑} → 𝑧 ∈ {∅})
3 velsn 3635 . . . . . . . . 9 (𝑧 ∈ {∅} ↔ 𝑧 = ∅)
42, 3sylib 122 . . . . . . . 8 (𝑧 ∈ {𝑥 ∈ {∅} ∣ 𝜑} → 𝑧 = ∅)
5 noel 3450 . . . . . . . . 9 ¬ 𝑦 ∈ ∅
6 eleq2 2257 . . . . . . . . 9 (𝑧 = ∅ → (𝑦𝑧𝑦 ∈ ∅))
75, 6mtbiri 676 . . . . . . . 8 (𝑧 = ∅ → ¬ 𝑦𝑧)
84, 7syl 14 . . . . . . 7 (𝑧 ∈ {𝑥 ∈ {∅} ∣ 𝜑} → ¬ 𝑦𝑧)
98adantl 277 . . . . . 6 ((𝑦𝑧𝑧 ∈ {𝑥 ∈ {∅} ∣ 𝜑}) → ¬ 𝑦𝑧)
101, 9pm2.21dd 621 . . . . 5 ((𝑦𝑧𝑧 ∈ {𝑥 ∈ {∅} ∣ 𝜑}) → 𝑦 ∈ {𝑥 ∈ {∅} ∣ 𝜑})
1110gen2 1461 . . . 4 𝑦𝑧((𝑦𝑧𝑧 ∈ {𝑥 ∈ {∅} ∣ 𝜑}) → 𝑦 ∈ {𝑥 ∈ {∅} ∣ 𝜑})
12 dftr2 4129 . . . 4 (Tr {𝑥 ∈ {∅} ∣ 𝜑} ↔ ∀𝑦𝑧((𝑦𝑧𝑧 ∈ {𝑥 ∈ {∅} ∣ 𝜑}) → 𝑦 ∈ {𝑥 ∈ {∅} ∣ 𝜑}))
1311, 12mpbir 146 . . 3 Tr {𝑥 ∈ {∅} ∣ 𝜑}
14 ssrab2 3264 . . 3 {𝑥 ∈ {∅} ∣ 𝜑} ⊆ {∅}
15 ord0 4422 . . . . 5 Ord ∅
16 ordsucim 4532 . . . . 5 (Ord ∅ → Ord suc ∅)
1715, 16ax-mp 5 . . . 4 Ord suc ∅
18 suc0 4442 . . . . 5 suc ∅ = {∅}
19 ordeq 4403 . . . . 5 (suc ∅ = {∅} → (Ord suc ∅ ↔ Ord {∅}))
2018, 19ax-mp 5 . . . 4 (Ord suc ∅ ↔ Ord {∅})
2117, 20mpbi 145 . . 3 Ord {∅}
22 trssord 4411 . . 3 ((Tr {𝑥 ∈ {∅} ∣ 𝜑} ∧ {𝑥 ∈ {∅} ∣ 𝜑} ⊆ {∅} ∧ Ord {∅}) → Ord {𝑥 ∈ {∅} ∣ 𝜑})
2313, 14, 21, 22mp3an 1348 . 2 Ord {𝑥 ∈ {∅} ∣ 𝜑}
24 p0ex 4217 . . . 4 {∅} ∈ V
2524rabex 4173 . . 3 {𝑥 ∈ {∅} ∣ 𝜑} ∈ V
2625elon 4405 . 2 ({𝑥 ∈ {∅} ∣ 𝜑} ∈ On ↔ Ord {𝑥 ∈ {∅} ∣ 𝜑})
2723, 26mpbir 146 1 {𝑥 ∈ {∅} ∣ 𝜑} ∈ On
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wb 105  wal 1362   = wceq 1364  wcel 2164  {crab 2476  wss 3153  c0 3446  {csn 3618  Tr wtr 4127  Ord word 4393  Oncon0 4394  suc csuc 4396
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-nul 4155  ax-pow 4203
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-uni 3836  df-tr 4128  df-iord 4397  df-on 4399  df-suc 4402
This theorem is referenced by:  ordtriexmid  4553  ontriexmidim  4554  ordtri2orexmid  4555  ontr2exmid  4557  onsucsssucexmid  4559  ordsoexmid  4594  0elsucexmid  4597  ordpwsucexmid  4602  unfiexmid  6974  exmidonfinlem  7253
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