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Mirrors > Home > ILE Home > Th. List > onintexmid | Unicode version |
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.) |
Ref | Expression |
---|---|
onintexmid.onint |
Ref | Expression |
---|---|
onintexmid |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prssi 3522 | . . . . . 6 | |
2 | prmg 3489 | . . . . . . 7 | |
3 | 2 | adantr 261 | . . . . . 6 |
4 | zfpair2 3945 | . . . . . . 7 | |
5 | sseq1 2966 | . . . . . . . . 9 | |
6 | eleq2 2101 | . . . . . . . . . 10 | |
7 | 6 | exbidv 1706 | . . . . . . . . 9 |
8 | 5, 7 | anbi12d 442 | . . . . . . . 8 |
9 | inteq 3618 | . . . . . . . . 9 | |
10 | id 19 | . . . . . . . . 9 | |
11 | 9, 10 | eleq12d 2108 | . . . . . . . 8 |
12 | 8, 11 | imbi12d 223 | . . . . . . 7 |
13 | onintexmid.onint | . . . . . . 7 | |
14 | 4, 12, 13 | vtocl 2608 | . . . . . 6 |
15 | 1, 3, 14 | syl2anc 391 | . . . . 5 |
16 | elpri 3398 | . . . . 5 | |
17 | 15, 16 | syl 14 | . . . 4 |
18 | incom 3129 | . . . . . . 7 | |
19 | 18 | eqeq1i 2047 | . . . . . 6 |
20 | dfss1 3141 | . . . . . 6 | |
21 | vex 2560 | . . . . . . . 8 | |
22 | vex 2560 | . . . . . . . 8 | |
23 | 21, 22 | intpr 3647 | . . . . . . 7 |
24 | 23 | eqeq1i 2047 | . . . . . 6 |
25 | 19, 20, 24 | 3bitr4ri 202 | . . . . 5 |
26 | 23 | eqeq1i 2047 | . . . . . 6 |
27 | dfss1 3141 | . . . . . 6 | |
28 | 26, 27 | bitr4i 176 | . . . . 5 |
29 | 25, 28 | orbi12i 681 | . . . 4 |
30 | 17, 29 | sylib 127 | . . 3 |
31 | 30 | rgen2a 2375 | . 2 |
32 | 31 | ordtri2or2exmid 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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