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Theorem ordtriexmidlem 4272
Description: Lemma for decidability and ordinals. The set {𝑥 ∈ {∅} ∣ 𝜑} is a way of connecting statements about ordinals (such as trichotomy in ordtriexmid 4274 or weak linearity in ordsoexmid 4313) 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 106 . . . . . 6 ((𝑦𝑧𝑧 ∈ {𝑥 ∈ {∅} ∣ 𝜑}) → 𝑦𝑧)
2 elrabi 2717 . . . . . . . . 9 (𝑧 ∈ {𝑥 ∈ {∅} ∣ 𝜑} → 𝑧 ∈ {∅})
3 velsn 3419 . . . . . . . . 9 (𝑧 ∈ {∅} ↔ 𝑧 = ∅)
42, 3sylib 131 . . . . . . . 8 (𝑧 ∈ {𝑥 ∈ {∅} ∣ 𝜑} → 𝑧 = ∅)
5 noel 3255 . . . . . . . . 9 ¬ 𝑦 ∈ ∅
6 eleq2 2117 . . . . . . . . 9 (𝑧 = ∅ → (𝑦𝑧𝑦 ∈ ∅))
75, 6mtbiri 610 . . . . . . . 8 (𝑧 = ∅ → ¬ 𝑦𝑧)
84, 7syl 14 . . . . . . 7 (𝑧 ∈ {𝑥 ∈ {∅} ∣ 𝜑} → ¬ 𝑦𝑧)
98adantl 266 . . . . . 6 ((𝑦𝑧𝑧 ∈ {𝑥 ∈ {∅} ∣ 𝜑}) → ¬ 𝑦𝑧)
101, 9pm2.21dd 560 . . . . 5 ((𝑦𝑧𝑧 ∈ {𝑥 ∈ {∅} ∣ 𝜑}) → 𝑦 ∈ {𝑥 ∈ {∅} ∣ 𝜑})
1110gen2 1355 . . . 4 𝑦𝑧((𝑦𝑧𝑧 ∈ {𝑥 ∈ {∅} ∣ 𝜑}) → 𝑦 ∈ {𝑥 ∈ {∅} ∣ 𝜑})
12 dftr2 3883 . . . 4 (Tr {𝑥 ∈ {∅} ∣ 𝜑} ↔ ∀𝑦𝑧((𝑦𝑧𝑧 ∈ {𝑥 ∈ {∅} ∣ 𝜑}) → 𝑦 ∈ {𝑥 ∈ {∅} ∣ 𝜑}))
1311, 12mpbir 138 . . 3 Tr {𝑥 ∈ {∅} ∣ 𝜑}
14 ssrab2 3052 . . 3 {𝑥 ∈ {∅} ∣ 𝜑} ⊆ {∅}
15 ord0 4155 . . . . 5 Ord ∅
16 ordsucim 4253 . . . . 5 (Ord ∅ → Ord suc ∅)
1715, 16ax-mp 7 . . . 4 Ord suc ∅
18 suc0 4175 . . . . 5 suc ∅ = {∅}
19 ordeq 4136 . . . . 5 (suc ∅ = {∅} → (Ord suc ∅ ↔ Ord {∅}))
2018, 19ax-mp 7 . . . 4 (Ord suc ∅ ↔ Ord {∅})
2117, 20mpbi 137 . . 3 Ord {∅}
22 trssord 4144 . . 3 ((Tr {𝑥 ∈ {∅} ∣ 𝜑} ∧ {𝑥 ∈ {∅} ∣ 𝜑} ⊆ {∅} ∧ Ord {∅}) → Ord {𝑥 ∈ {∅} ∣ 𝜑})
2313, 14, 21, 22mp3an 1243 . 2 Ord {𝑥 ∈ {∅} ∣ 𝜑}
24 p0ex 3966 . . . 4 {∅} ∈ V
2524rabex 3928 . . 3 {𝑥 ∈ {∅} ∣ 𝜑} ∈ V
2625elon 4138 . 2 ({𝑥 ∈ {∅} ∣ 𝜑} ∈ On ↔ Ord {𝑥 ∈ {∅} ∣ 𝜑})
2723, 26mpbir 138 1 {𝑥 ∈ {∅} ∣ 𝜑} ∈ On
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 101  wb 102  wal 1257   = wceq 1259  wcel 1409  {crab 2327  wss 2944  c0 3251  {csn 3402  Tr wtr 3881  Ord word 4126  Oncon0 4127  suc csuc 4129
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-nul 3910  ax-pow 3954
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-rab 2332  df-v 2576  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-nul 3252  df-pw 3388  df-sn 3408  df-uni 3608  df-tr 3882  df-iord 4130  df-on 4132  df-suc 4135
This theorem is referenced by:  ordtriexmid  4274  ordtri2orexmid  4275  ontr2exmid  4277  onsucsssucexmid  4279  ordsoexmid  4313  0elsucexmid  4316  ordpwsucexmid  4321
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