ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ordtriexmidlem GIF version

Theorem ordtriexmidlem 4403
Description: Lemma for decidability and ordinals. The set {𝑥 ∈ {∅} ∣ 𝜑} is a way of connecting statements about ordinals (such as trichotomy in ordtriexmid 4405 or weak linearity in ordsoexmid 4445) 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 108 . . . . . 6 ((𝑦𝑧𝑧 ∈ {𝑥 ∈ {∅} ∣ 𝜑}) → 𝑦𝑧)
2 elrabi 2808 . . . . . . . . 9 (𝑧 ∈ {𝑥 ∈ {∅} ∣ 𝜑} → 𝑧 ∈ {∅})
3 velsn 3512 . . . . . . . . 9 (𝑧 ∈ {∅} ↔ 𝑧 = ∅)
42, 3sylib 121 . . . . . . . 8 (𝑧 ∈ {𝑥 ∈ {∅} ∣ 𝜑} → 𝑧 = ∅)
5 noel 3335 . . . . . . . . 9 ¬ 𝑦 ∈ ∅
6 eleq2 2179 . . . . . . . . 9 (𝑧 = ∅ → (𝑦𝑧𝑦 ∈ ∅))
75, 6mtbiri 647 . . . . . . . 8 (𝑧 = ∅ → ¬ 𝑦𝑧)
84, 7syl 14 . . . . . . 7 (𝑧 ∈ {𝑥 ∈ {∅} ∣ 𝜑} → ¬ 𝑦𝑧)
98adantl 273 . . . . . 6 ((𝑦𝑧𝑧 ∈ {𝑥 ∈ {∅} ∣ 𝜑}) → ¬ 𝑦𝑧)
101, 9pm2.21dd 592 . . . . 5 ((𝑦𝑧𝑧 ∈ {𝑥 ∈ {∅} ∣ 𝜑}) → 𝑦 ∈ {𝑥 ∈ {∅} ∣ 𝜑})
1110gen2 1409 . . . 4 𝑦𝑧((𝑦𝑧𝑧 ∈ {𝑥 ∈ {∅} ∣ 𝜑}) → 𝑦 ∈ {𝑥 ∈ {∅} ∣ 𝜑})
12 dftr2 3996 . . . 4 (Tr {𝑥 ∈ {∅} ∣ 𝜑} ↔ ∀𝑦𝑧((𝑦𝑧𝑧 ∈ {𝑥 ∈ {∅} ∣ 𝜑}) → 𝑦 ∈ {𝑥 ∈ {∅} ∣ 𝜑}))
1311, 12mpbir 145 . . 3 Tr {𝑥 ∈ {∅} ∣ 𝜑}
14 ssrab2 3150 . . 3 {𝑥 ∈ {∅} ∣ 𝜑} ⊆ {∅}
15 ord0 4281 . . . . 5 Ord ∅
16 ordsucim 4384 . . . . 5 (Ord ∅ → Ord suc ∅)
1715, 16ax-mp 5 . . . 4 Ord suc ∅
18 suc0 4301 . . . . 5 suc ∅ = {∅}
19 ordeq 4262 . . . . 5 (suc ∅ = {∅} → (Ord suc ∅ ↔ Ord {∅}))
2018, 19ax-mp 5 . . . 4 (Ord suc ∅ ↔ Ord {∅})
2117, 20mpbi 144 . . 3 Ord {∅}
22 trssord 4270 . . 3 ((Tr {𝑥 ∈ {∅} ∣ 𝜑} ∧ {𝑥 ∈ {∅} ∣ 𝜑} ⊆ {∅} ∧ Ord {∅}) → Ord {𝑥 ∈ {∅} ∣ 𝜑})
2313, 14, 21, 22mp3an 1298 . 2 Ord {𝑥 ∈ {∅} ∣ 𝜑}
24 p0ex 4080 . . . 4 {∅} ∈ V
2524rabex 4040 . . 3 {𝑥 ∈ {∅} ∣ 𝜑} ∈ V
2625elon 4264 . 2 ({𝑥 ∈ {∅} ∣ 𝜑} ∈ On ↔ Ord {𝑥 ∈ {∅} ∣ 𝜑})
2723, 26mpbir 145 1 {𝑥 ∈ {∅} ∣ 𝜑} ∈ On
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104  wal 1312   = wceq 1314  wcel 1463  {crab 2395  wss 3039  c0 3331  {csn 3495  Tr wtr 3994  Ord word 4252  Oncon0 4253  suc csuc 4255
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-nul 4022  ax-pow 4066
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-rab 2400  df-v 2660  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-uni 3705  df-tr 3995  df-iord 4256  df-on 4258  df-suc 4261
This theorem is referenced by:  ordtriexmid  4405  ordtri2orexmid  4406  ontr2exmid  4408  onsucsssucexmid  4410  ordsoexmid  4445  0elsucexmid  4448  ordpwsucexmid  4453  unfiexmid  6772
  Copyright terms: Public domain W3C validator