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Theorem ordtriexmidlem 4192
Description: Lemma for decidability and ordinals. The set {x {∅} ∣ φ} is a way of connecting statements about ordinals (such as trichotomy in ordtriexmid 4194 or weak linearity in ordsoexmid 4224) 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 ax-ia1 99 . . . . . 6 ((y z z {x {∅} ∣ φ}) → y z)
2 elrabi 2672 . . . . . . . . 9 (z {x {∅} ∣ φ} → z {∅})
3 elsn 3365 . . . . . . . . 9 (z {∅} ↔ z = ∅)
42, 3sylib 127 . . . . . . . 8 (z {x {∅} ∣ φ} → z = ∅)
5 noel 3205 . . . . . . . . 9 ¬ y
6 eleq2 2083 . . . . . . . . 9 (z = ∅ → (y zy ∅))
75, 6mtbiri 587 . . . . . . . 8 (z = ∅ → ¬ y z)
84, 7syl 14 . . . . . . 7 (z {x {∅} ∣ φ} → ¬ y z)
98adantl 262 . . . . . 6 ((y z z {x {∅} ∣ φ}) → ¬ y z)
101, 9pm2.21dd 538 . . . . 5 ((y z z {x {∅} ∣ φ}) → y {x {∅} ∣ φ})
1110gen2 1319 . . . 4 yz((y z z {x {∅} ∣ φ}) → y {x {∅} ∣ φ})
12 dftr2 3830 . . . 4 (Tr {x {∅} ∣ φ} ↔ yz((y z z {x {∅} ∣ φ}) → y {x {∅} ∣ φ}))
1311, 12mpbir 134 . . 3 Tr {x {∅} ∣ φ}
14 ssrab2 3002 . . 3 {x {∅} ∣ φ} ⊆ {∅}
15 ord0 4077 . . . . 5 Ord ∅
16 ordsucim 4176 . . . . 5 (Ord ∅ → Ord suc ∅)
1715, 16ax-mp 7 . . . 4 Ord suc ∅
18 suc0 4097 . . . . 5 suc ∅ = {∅}
19 ordeq 4058 . . . . 5 (suc ∅ = {∅} → (Ord suc ∅ ↔ Ord {∅}))
2018, 19ax-mp 7 . . . 4 (Ord suc ∅ ↔ Ord {∅})
2117, 20mpbi 133 . . 3 Ord {∅}
22 trssord 4066 . . 3 ((Tr {x {∅} ∣ φ} {x {∅} ∣ φ} ⊆ {∅} Ord {∅}) → Ord {x {∅} ∣ φ})
2313, 14, 21, 22mp3an 1217 . 2 Ord {x {∅} ∣ φ}
24 p0ex 3913 . . . 4 {∅} V
2524rabex 3875 . . 3 {x {∅} ∣ φ} V
2625elon 4060 . 2 ({x {∅} ∣ φ} On ↔ Ord {x {∅} ∣ φ})
2723, 26mpbir 134 1 {x {∅} ∣ φ} On
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97  wb 98  wal 1226   = wceq 1228   wcel 1374  {crab 2288  wss 2894  c0 3201  {csn 3350  Tr wtr 3828  Ord word 4048  Oncon0 4049  suc csuc 4051
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-nul 3857  ax-pow 3901
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-rab 2293  df-v 2537  df-dif 2897  df-un 2899  df-in 2901  df-ss 2908  df-nul 3202  df-pw 3336  df-sn 3356  df-uni 3555  df-tr 3829  df-iord 4052  df-on 4054  df-suc 4057
This theorem is referenced by:  ordtriexmid  4194  ordtri2orexmid  4195  onsucsssucexmid  4196  ordsoexmid  4224  ordpwsucexmid  4230
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