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

Theorem ordtriexmidlem 4515
Description: Lemma for decidability and ordinals. The set {𝑥 ∈ {∅} ∣ 𝜑} is a way of connecting statements about ordinals (such as trichotomy in ordtriexmid 4517 or weak linearity in ordsoexmid 4558) 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 2890 . . . . . . . . 9 (𝑧 ∈ {𝑥 ∈ {∅} ∣ 𝜑} → 𝑧 ∈ {∅})
3 velsn 3608 . . . . . . . . 9 (𝑧 ∈ {∅} ↔ 𝑧 = ∅)
42, 3sylib 122 . . . . . . . 8 (𝑧 ∈ {𝑥 ∈ {∅} ∣ 𝜑} → 𝑧 = ∅)
5 noel 3426 . . . . . . . . 9 ¬ 𝑦 ∈ ∅
6 eleq2 2241 . . . . . . . . 9 (𝑧 = ∅ → (𝑦𝑧𝑦 ∈ ∅))
75, 6mtbiri 675 . . . . . . . 8 (𝑧 = ∅ → ¬ 𝑦𝑧)
84, 7syl 14 . . . . . . 7 (𝑧 ∈ {𝑥 ∈ {∅} ∣ 𝜑} → ¬ 𝑦𝑧)
98adantl 277 . . . . . 6 ((𝑦𝑧𝑧 ∈ {𝑥 ∈ {∅} ∣ 𝜑}) → ¬ 𝑦𝑧)
101, 9pm2.21dd 620 . . . . 5 ((𝑦𝑧𝑧 ∈ {𝑥 ∈ {∅} ∣ 𝜑}) → 𝑦 ∈ {𝑥 ∈ {∅} ∣ 𝜑})
1110gen2 1450 . . . 4 𝑦𝑧((𝑦𝑧𝑧 ∈ {𝑥 ∈ {∅} ∣ 𝜑}) → 𝑦 ∈ {𝑥 ∈ {∅} ∣ 𝜑})
12 dftr2 4100 . . . 4 (Tr {𝑥 ∈ {∅} ∣ 𝜑} ↔ ∀𝑦𝑧((𝑦𝑧𝑧 ∈ {𝑥 ∈ {∅} ∣ 𝜑}) → 𝑦 ∈ {𝑥 ∈ {∅} ∣ 𝜑}))
1311, 12mpbir 146 . . 3 Tr {𝑥 ∈ {∅} ∣ 𝜑}
14 ssrab2 3240 . . 3 {𝑥 ∈ {∅} ∣ 𝜑} ⊆ {∅}
15 ord0 4388 . . . . 5 Ord ∅
16 ordsucim 4496 . . . . 5 (Ord ∅ → Ord suc ∅)
1715, 16ax-mp 5 . . . 4 Ord suc ∅
18 suc0 4408 . . . . 5 suc ∅ = {∅}
19 ordeq 4369 . . . . 5 (suc ∅ = {∅} → (Ord suc ∅ ↔ Ord {∅}))
2018, 19ax-mp 5 . . . 4 (Ord suc ∅ ↔ Ord {∅})
2117, 20mpbi 145 . . 3 Ord {∅}
22 trssord 4377 . . 3 ((Tr {𝑥 ∈ {∅} ∣ 𝜑} ∧ {𝑥 ∈ {∅} ∣ 𝜑} ⊆ {∅} ∧ Ord {∅}) → Ord {𝑥 ∈ {∅} ∣ 𝜑})
2313, 14, 21, 22mp3an 1337 . 2 Ord {𝑥 ∈ {∅} ∣ 𝜑}
24 p0ex 4185 . . . 4 {∅} ∈ V
2524rabex 4144 . . 3 {𝑥 ∈ {∅} ∣ 𝜑} ∈ V
2625elon 4371 . 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 1351   = wceq 1353  wcel 2148  {crab 2459  wss 3129  c0 3422  {csn 3591  Tr wtr 4098  Ord word 4359  Oncon0 4360  suc csuc 4362
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-nul 4126  ax-pow 4171
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-uni 3808  df-tr 4099  df-iord 4363  df-on 4365  df-suc 4368
This theorem is referenced by:  ordtriexmid  4517  ontriexmidim  4518  ordtri2orexmid  4519  ontr2exmid  4521  onsucsssucexmid  4523  ordsoexmid  4558  0elsucexmid  4561  ordpwsucexmid  4566  unfiexmid  6911  exmidonfinlem  7186
  Copyright terms: Public domain W3C validator