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Theorem ordtriexmidlem 4477
 Description: Lemma for decidability and ordinals. The set {𝑥 ∈ {∅} ∣ 𝜑} is a way of connecting statements about ordinals (such as trichotomy in ordtriexmid 4479 or weak linearity in ordsoexmid 4520) 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 2865 . . . . . . . . 9 (𝑧 ∈ {𝑥 ∈ {∅} ∣ 𝜑} → 𝑧 ∈ {∅})
3 velsn 3577 . . . . . . . . 9 (𝑧 ∈ {∅} ↔ 𝑧 = ∅)
42, 3sylib 121 . . . . . . . 8 (𝑧 ∈ {𝑥 ∈ {∅} ∣ 𝜑} → 𝑧 = ∅)
5 noel 3398 . . . . . . . . 9 ¬ 𝑦 ∈ ∅
6 eleq2 2221 . . . . . . . . 9 (𝑧 = ∅ → (𝑦𝑧𝑦 ∈ ∅))
75, 6mtbiri 665 . . . . . . . 8 (𝑧 = ∅ → ¬ 𝑦𝑧)
84, 7syl 14 . . . . . . 7 (𝑧 ∈ {𝑥 ∈ {∅} ∣ 𝜑} → ¬ 𝑦𝑧)
98adantl 275 . . . . . 6 ((𝑦𝑧𝑧 ∈ {𝑥 ∈ {∅} ∣ 𝜑}) → ¬ 𝑦𝑧)
101, 9pm2.21dd 610 . . . . 5 ((𝑦𝑧𝑧 ∈ {𝑥 ∈ {∅} ∣ 𝜑}) → 𝑦 ∈ {𝑥 ∈ {∅} ∣ 𝜑})
1110gen2 1430 . . . 4 𝑦𝑧((𝑦𝑧𝑧 ∈ {𝑥 ∈ {∅} ∣ 𝜑}) → 𝑦 ∈ {𝑥 ∈ {∅} ∣ 𝜑})
12 dftr2 4064 . . . 4 (Tr {𝑥 ∈ {∅} ∣ 𝜑} ↔ ∀𝑦𝑧((𝑦𝑧𝑧 ∈ {𝑥 ∈ {∅} ∣ 𝜑}) → 𝑦 ∈ {𝑥 ∈ {∅} ∣ 𝜑}))
1311, 12mpbir 145 . . 3 Tr {𝑥 ∈ {∅} ∣ 𝜑}
14 ssrab2 3213 . . 3 {𝑥 ∈ {∅} ∣ 𝜑} ⊆ {∅}
15 ord0 4351 . . . . 5 Ord ∅
16 ordsucim 4458 . . . . 5 (Ord ∅ → Ord suc ∅)
1715, 16ax-mp 5 . . . 4 Ord suc ∅
18 suc0 4371 . . . . 5 suc ∅ = {∅}
19 ordeq 4332 . . . . 5 (suc ∅ = {∅} → (Ord suc ∅ ↔ Ord {∅}))
2018, 19ax-mp 5 . . . 4 (Ord suc ∅ ↔ Ord {∅})
2117, 20mpbi 144 . . 3 Ord {∅}
22 trssord 4340 . . 3 ((Tr {𝑥 ∈ {∅} ∣ 𝜑} ∧ {𝑥 ∈ {∅} ∣ 𝜑} ⊆ {∅} ∧ Ord {∅}) → Ord {𝑥 ∈ {∅} ∣ 𝜑})
2313, 14, 21, 22mp3an 1319 . 2 Ord {𝑥 ∈ {∅} ∣ 𝜑}
24 p0ex 4149 . . . 4 {∅} ∈ V
2524rabex 4108 . . 3 {𝑥 ∈ {∅} ∣ 𝜑} ∈ V
2625elon 4334 . 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 1333   = wceq 1335   ∈ wcel 2128  {crab 2439   ⊆ wss 3102  ∅c0 3394  {csn 3560  Tr wtr 4062  Ord word 4322  Oncon0 4323  suc csuc 4325 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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-nul 4090  ax-pow 4135 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-uni 3773  df-tr 4063  df-iord 4326  df-on 4328  df-suc 4331 This theorem is referenced by:  ordtriexmid  4479  ontriexmidim  4480  ordtri2orexmid  4481  ontr2exmid  4483  onsucsssucexmid  4485  ordsoexmid  4520  0elsucexmid  4523  ordpwsucexmid  4528  unfiexmid  6859  exmidonfinlem  7122
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