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Theorem onsucelsucexmidlem 4254
 Description: Lemma for onsucelsucexmid 4255. The set appears as in the proof of Theorem 1.3 in [Bauer] p. 483 (see acexmidlema 5503), and similar sets also appear in other proofs that various propositions imply excluded middle, for example in ordtriexmidlem 4245. (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 481 . . . . . . . 8
2 noel 3228 . . . . . . . . . 10
3 eleq2 2101 . . . . . . . . . 10
42, 3mtbiri 600 . . . . . . . . 9
54adantl 262 . . . . . . . 8
61, 5pm2.21dd 550 . . . . . . 7
76ex 108 . . . . . 6
8 eleq2 2101 . . . . . . . . . . 11
98biimpac 282 . . . . . . . . . 10
10 velsn 3392 . . . . . . . . . 10
119, 10sylib 127 . . . . . . . . 9
12 onsucelsucexmidlem1 4253 . . . . . . . . 9
1311, 12syl6eqel 2128 . . . . . . . 8
1413ex 108 . . . . . . 7
1514adantr 261 . . . . . 6
16 elrabi 2695 . . . . . . . 8
17 vex 2560 . . . . . . . . 9
1817elpr 3396 . . . . . . . 8
1916, 18sylib 127 . . . . . . 7
2019adantl 262 . . . . . 6
217, 15, 20mpjaod 638 . . . . 5
2221gen2 1339 . . . 4
23 dftr2 3856 . . . 4
2422, 23mpbir 134 . . 3
25 ssrab2 3025 . . 3
26 2ordpr 4249 . . 3
27 trssord 4117 . . 3
2824, 25, 26, 27mp3an 1232 . 2
29 pp0ex 3940 . . . 4
3029rabex 3901 . . 3
3130elon 4111 . 2
3228, 31mpbir 134 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 97   wo 629  wal 1241   wceq 1243   wcel 1393  crab 2310   wss 2917  c0 3224  csn 3375  cpr 3376   wtr 3854   word 4099  con0 4100 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 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-nul 3883  ax-pow 3927 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-uni 3581  df-tr 3855  df-iord 4103  df-on 4105  df-suc 4108 This theorem is referenced by:  onsucelsucexmid  4255  acexmidlemcase  5507  acexmidlemv  5510
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