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Theorem onintexmid 4297
 Description: If the intersection (infimum) of an inhabited class of ordinal numbers belongs to the class, excluded middle follows. The hypothesis would be provable given excluded middle. (Contributed by Mario Carneiro and Jim Kingdon, 29-Aug-2021.)
Hypothesis
Ref Expression
onintexmid.onint
Assertion
Ref Expression
onintexmid
Distinct variable group:   ,
Allowed substitution hints:   (,)

Proof of Theorem onintexmid
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prssi 3522 . . . . . 6
2 prmg 3489 . . . . . . 7
32adantr 261 . . . . . 6
4 zfpair2 3945 . . . . . . 7
5 sseq1 2966 . . . . . . . . 9
6 eleq2 2101 . . . . . . . . . 10
76exbidv 1706 . . . . . . . . 9
85, 7anbi12d 442 . . . . . . . 8
9 inteq 3618 . . . . . . . . 9
10 id 19 . . . . . . . . 9
119, 10eleq12d 2108 . . . . . . . 8
128, 11imbi12d 223 . . . . . . 7
13 onintexmid.onint . . . . . . 7
144, 12, 13vtocl 2608 . . . . . 6
151, 3, 14syl2anc 391 . . . . 5
16 elpri 3398 . . . . 5
1715, 16syl 14 . . . 4
18 incom 3129 . . . . . . 7
1918eqeq1i 2047 . . . . . 6
20 dfss1 3141 . . . . . 6
21 vex 2560 . . . . . . . 8
22 vex 2560 . . . . . . . 8
2321, 22intpr 3647 . . . . . . 7
2423eqeq1i 2047 . . . . . 6
2519, 20, 243bitr4ri 202 . . . . 5
2623eqeq1i 2047 . . . . . 6
27 dfss1 3141 . . . . . 6
2826, 27bitr4i 176 . . . . 5
2925, 28orbi12i 681 . . . 4
3017, 29sylib 127 . . 3
3130rgen2a 2375 . 2
3231ordtri2or2exmid 4296 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 97   wo 629   wceq 1243  wex 1381   wcel 1393   cin 2916   wss 2917  cpr 3376  cint 3615  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-13 1404  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  ax-pr 3944  ax-un 4170  ax-setind 4262 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-ne 2206  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-int 3616  df-tr 3855  df-iord 4103  df-on 4105  df-suc 4108 This theorem is referenced by: (None)
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