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Theorem onsucelsucexmidlem 4439
 Description: Lemma for onsucelsucexmid 4440. The set appears as in the proof of Theorem 1.3 in [Bauer] p. 483 (see acexmidlema 5758), and similar sets also appear in other proofs that various propositions imply excluded middle, for example in ordtriexmidlem 4430. (Contributed by Jim Kingdon, 2-Aug-2019.)
Assertion
Ref Expression
onsucelsucexmidlem
Distinct variable group:   ,

Proof of Theorem onsucelsucexmidlem
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpll 518 . . . . . . . 8
2 noel 3362 . . . . . . . . . 10
3 eleq2 2201 . . . . . . . . . 10
42, 3mtbiri 664 . . . . . . . . 9
54adantl 275 . . . . . . . 8
61, 5pm2.21dd 609 . . . . . . 7
76ex 114 . . . . . 6
8 eleq2 2201 . . . . . . . . . . 11
98biimpac 296 . . . . . . . . . 10
10 velsn 3539 . . . . . . . . . 10
119, 10sylib 121 . . . . . . . . 9
12 onsucelsucexmidlem1 4438 . . . . . . . . 9
1311, 12eqeltrdi 2228 . . . . . . . 8
1413ex 114 . . . . . . 7
1514adantr 274 . . . . . 6
16 elrabi 2832 . . . . . . . 8
17 vex 2684 . . . . . . . . 9
1817elpr 3543 . . . . . . . 8
1916, 18sylib 121 . . . . . . 7
2019adantl 275 . . . . . 6
217, 15, 20mpjaod 707 . . . . 5
2221gen2 1426 . . . 4
23 dftr2 4023 . . . 4
2422, 23mpbir 145 . . 3
25 ssrab2 3177 . . 3
26 2ordpr 4434 . . 3
27 trssord 4297 . . 3
2824, 25, 26, 27mp3an 1315 . 2
29 pp0ex 4108 . . . 4
3029rabex 4067 . . 3
3130elon 4291 . 2
3228, 31mpbir 145 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 103   wo 697  wal 1329   wceq 1331   wcel 1480  crab 2418   wss 3066  c0 3358  csn 3522  cpr 3523   wtr 4021   word 4279  con0 4280 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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-nul 4049  ax-pow 4093 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-rab 2423  df-v 2683  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-uni 3732  df-tr 4022  df-iord 4283  df-on 4285  df-suc 4288 This theorem is referenced by:  onsucelsucexmid  4440  acexmidlemcase  5762  acexmidlemv  5765
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