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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 3716 | . . . . . 6 | |
2 | prmg 3682 | . . . . . . 7 | |
3 | 2 | adantr 274 | . . . . . 6 |
4 | zfpair2 4172 | . . . . . . 7 | |
5 | sseq1 3151 | . . . . . . . . 9 | |
6 | eleq2 2221 | . . . . . . . . . 10 | |
7 | 6 | exbidv 1805 | . . . . . . . . 9 |
8 | 5, 7 | anbi12d 465 | . . . . . . . 8 |
9 | inteq 3812 | . . . . . . . . 9 | |
10 | id 19 | . . . . . . . . 9 | |
11 | 9, 10 | eleq12d 2228 | . . . . . . . 8 |
12 | 8, 11 | imbi12d 233 | . . . . . . 7 |
13 | onintexmid.onint | . . . . . . 7 | |
14 | 4, 12, 13 | vtocl 2766 | . . . . . 6 |
15 | 1, 3, 14 | syl2anc 409 | . . . . 5 |
16 | elpri 3584 | . . . . 5 | |
17 | 15, 16 | syl 14 | . . . 4 |
18 | incom 3300 | . . . . . . 7 | |
19 | 18 | eqeq1i 2165 | . . . . . 6 |
20 | dfss1 3312 | . . . . . 6 | |
21 | vex 2715 | . . . . . . . 8 | |
22 | vex 2715 | . . . . . . . 8 | |
23 | 21, 22 | intpr 3841 | . . . . . . 7 |
24 | 23 | eqeq1i 2165 | . . . . . 6 |
25 | 19, 20, 24 | 3bitr4ri 212 | . . . . 5 |
26 | 23 | eqeq1i 2165 | . . . . . 6 |
27 | dfss1 3312 | . . . . . 6 | |
28 | 26, 27 | bitr4i 186 | . . . . 5 |
29 | 25, 28 | orbi12i 754 | . . . 4 |
30 | 17, 29 | sylib 121 | . . 3 |
31 | 30 | rgen2a 2511 | . 2 |
32 | 31 | ordtri2or2exmid 4532 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wo 698 wceq 1335 wex 1472 wcel 2128 cin 3101 wss 3102 cpr 3562 cint 3809 con0 4325 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4084 ax-nul 4092 ax-pow 4137 ax-pr 4171 ax-un 4395 ax-setind 4498 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-ral 2440 df-rex 2441 df-rab 2444 df-v 2714 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3396 df-pw 3546 df-sn 3567 df-pr 3568 df-uni 3775 df-int 3810 df-tr 4065 df-iord 4328 df-on 4330 df-suc 4333 |
This theorem is referenced by: (None) |
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