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Theorem ordtriexmidlem 4271
 Description: Lemma for decidability and ordinals. The set is a way of connecting statements about ordinals (such as trichotomy in ordtriexmid 4273 or weak linearity in ordsoexmid 4313) with a proposition . Our lemma states that it is an ordinal number. (Contributed by Jim Kingdon, 28-Jan-2019.)
Assertion
Ref Expression
ordtriexmidlem

Proof of Theorem ordtriexmidlem
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 107 . . . . . 6
2 elrabi 2747 . . . . . . . . 9
3 velsn 3423 . . . . . . . . 9
42, 3sylib 120 . . . . . . . 8
5 noel 3262 . . . . . . . . 9
6 eleq2 2143 . . . . . . . . 9
75, 6mtbiri 633 . . . . . . . 8
84, 7syl 14 . . . . . . 7
98adantl 271 . . . . . 6
101, 9pm2.21dd 583 . . . . 5
1110gen2 1380 . . . 4
12 dftr2 3885 . . . 4
1311, 12mpbir 144 . . 3
14 ssrab2 3080 . . 3
15 ord0 4154 . . . . 5
16 ordsucim 4252 . . . . 5
1715, 16ax-mp 7 . . . 4
18 suc0 4174 . . . . 5
19 ordeq 4135 . . . . 5
2018, 19ax-mp 7 . . . 4
2117, 20mpbi 143 . . 3
22 trssord 4143 . . 3
2313, 14, 21, 22mp3an 1269 . 2
24 p0ex 3967 . . . 4
2524rabex 3930 . . 3
2625elon 4137 . 2
2723, 26mpbir 144 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 102   wb 103  wal 1283   wceq 1285   wcel 1434  crab 2353   wss 2974  c0 3258  csn 3406   wtr 3883   word 4125  con0 4126   csuc 4128 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-nul 3912  ax-pow 3956 This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-rab 2358  df-v 2604  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-pw 3392  df-sn 3412  df-uni 3610  df-tr 3884  df-iord 4129  df-on 4131  df-suc 4134 This theorem is referenced by:  ordtriexmid  4273  ordtri2orexmid  4274  ontr2exmid  4276  onsucsssucexmid  4278  ordsoexmid  4313  0elsucexmid  4316  ordpwsucexmid  4321  unfiexmid  6438
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