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Theorem reg2exmid 4261
 Description: If any inhabited set has a minimal element (when expressed by ), excluded middle follows. (Contributed by Jim Kingdon, 2-Oct-2021.)
Hypothesis
Ref Expression
reg2exmid.1
Assertion
Ref Expression
reg2exmid
Distinct variable groups:   ,,   ,,,

Proof of Theorem reg2exmid
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 eqid 2040 . . . 4
21regexmidlemm 4257 . . 3
3 reg2exmid.1 . . . 4
4 pp0ex 3940 . . . . . 6
54rabex 3901 . . . . 5
6 eleq2 2101 . . . . . . 7
76exbidv 1706 . . . . . 6
8 raleq 2505 . . . . . . 7
98rexeqbi1dv 2514 . . . . . 6
107, 9imbi12d 223 . . . . 5
115, 10spcv 2646 . . . 4
123, 11ax-mp 7 . . 3
132, 12ax-mp 7 . 2
141reg2exmidlema 4259 . 2
1513, 14ax-mp 7 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 97   wo 629  wal 1241   wceq 1243  wex 1381   wcel 1393  wral 2306  wrex 2307  crab 2310   wss 2917  c0 3224  csn 3375  cpr 3376 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-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 This theorem is referenced by: (None)
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