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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 3543 | . . . . . 6 | |
2 | prmg 3511 | . . . . . . 7 | |
3 | 2 | adantr 270 | . . . . . 6 |
4 | zfpair2 3965 | . . . . . . 7 | |
5 | sseq1 3020 | . . . . . . . . 9 | |
6 | eleq2 2142 | . . . . . . . . . 10 | |
7 | 6 | exbidv 1746 | . . . . . . . . 9 |
8 | 5, 7 | anbi12d 456 | . . . . . . . 8 |
9 | inteq 3639 | . . . . . . . . 9 | |
10 | id 19 | . . . . . . . . 9 | |
11 | 9, 10 | eleq12d 2149 | . . . . . . . 8 |
12 | 8, 11 | imbi12d 232 | . . . . . . 7 |
13 | onintexmid.onint | . . . . . . 7 | |
14 | 4, 12, 13 | vtocl 2653 | . . . . . 6 |
15 | 1, 3, 14 | syl2anc 403 | . . . . 5 |
16 | elpri 3421 | . . . . 5 | |
17 | 15, 16 | syl 14 | . . . 4 |
18 | incom 3158 | . . . . . . 7 | |
19 | 18 | eqeq1i 2088 | . . . . . 6 |
20 | dfss1 3170 | . . . . . 6 | |
21 | vex 2604 | . . . . . . . 8 | |
22 | vex 2604 | . . . . . . . 8 | |
23 | 21, 22 | intpr 3668 | . . . . . . 7 |
24 | 23 | eqeq1i 2088 | . . . . . 6 |
25 | 19, 20, 24 | 3bitr4ri 211 | . . . . 5 |
26 | 23 | eqeq1i 2088 | . . . . . 6 |
27 | dfss1 3170 | . . . . . 6 | |
28 | 26, 27 | bitr4i 185 | . . . . 5 |
29 | 25, 28 | orbi12i 713 | . . . 4 |
30 | 17, 29 | sylib 120 | . . 3 |
31 | 30 | rgen2a 2417 | . 2 |
32 | 31 | ordtri2or2exmid 4314 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 102 wo 661 wceq 1284 wex 1421 wcel 1433 cin 2972 wss 2973 cpr 3399 cint 3636 con0 4118 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rex 2354 df-rab 2357 df-v 2603 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-uni 3602 df-int 3637 df-tr 3876 df-iord 4121 df-on 4123 df-suc 4126 |
This theorem is referenced by: (None) |
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