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Mirrors > Home > ILE Home > Th. List > onsucelsucexmid | Unicode version |
Description: The converse of onsucelsucr 4479 implies excluded middle. On the other hand, if is constrained to be a natural number, instead of an arbitrary ordinal, then the converse of onsucelsucr 4479 does hold, as seen at nnsucelsuc 6450. (Contributed by Jim Kingdon, 2-Aug-2019.) |
Ref | Expression |
---|---|
onsucelsucexmid.1 |
Ref | Expression |
---|---|
onsucelsucexmid |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | onsucelsucexmidlem1 4499 | . . . 4 | |
2 | 0elon 4364 | . . . . . 6 | |
3 | onsucelsucexmidlem 4500 | . . . . . 6 | |
4 | 2, 3 | pm3.2i 270 | . . . . 5 |
5 | onsucelsucexmid.1 | . . . . 5 | |
6 | eleq1 2227 | . . . . . . 7 | |
7 | suceq 4374 | . . . . . . . 8 | |
8 | 7 | eleq1d 2233 | . . . . . . 7 |
9 | 6, 8 | imbi12d 233 | . . . . . 6 |
10 | eleq2 2228 | . . . . . . 7 | |
11 | suceq 4374 | . . . . . . . 8 | |
12 | 11 | eleq2d 2234 | . . . . . . 7 |
13 | 10, 12 | imbi12d 233 | . . . . . 6 |
14 | 9, 13 | rspc2va 2839 | . . . . 5 |
15 | 4, 5, 14 | mp2an 423 | . . . 4 |
16 | 1, 15 | ax-mp 5 | . . 3 |
17 | elsuci 4375 | . . 3 | |
18 | 16, 17 | ax-mp 5 | . 2 |
19 | suc0 4383 | . . . . . 6 | |
20 | p0ex 4161 | . . . . . . 7 | |
21 | 20 | prid2 3677 | . . . . . 6 |
22 | 19, 21 | eqeltri 2237 | . . . . 5 |
23 | eqeq1 2171 | . . . . . . 7 | |
24 | 23 | orbi1d 781 | . . . . . 6 |
25 | 24 | elrab3 2878 | . . . . 5 |
26 | 22, 25 | ax-mp 5 | . . . 4 |
27 | 0ex 4103 | . . . . . . 7 | |
28 | nsuceq0g 4390 | . . . . . . 7 | |
29 | 27, 28 | ax-mp 5 | . . . . . 6 |
30 | df-ne 2335 | . . . . . 6 | |
31 | 29, 30 | mpbi 144 | . . . . 5 |
32 | pm2.53 712 | . . . . 5 | |
33 | 31, 32 | mpi 15 | . . . 4 |
34 | 26, 33 | sylbi 120 | . . 3 |
35 | 19 | eqeq1i 2172 | . . . . 5 |
36 | 19 | eqeq1i 2172 | . . . . . . . 8 |
37 | 31, 36 | mtbi 660 | . . . . . . 7 |
38 | 20 | elsn 3586 | . . . . . . 7 |
39 | 37, 38 | mtbir 661 | . . . . . 6 |
40 | eleq2 2228 | . . . . . 6 | |
41 | 39, 40 | mtbii 664 | . . . . 5 |
42 | 35, 41 | sylbi 120 | . . . 4 |
43 | olc 701 | . . . . 5 | |
44 | eqeq1 2171 | . . . . . . . 8 | |
45 | 44 | orbi1d 781 | . . . . . . 7 |
46 | 45 | elrab3 2878 | . . . . . 6 |
47 | 21, 46 | ax-mp 5 | . . . . 5 |
48 | 43, 47 | sylibr 133 | . . . 4 |
49 | 42, 48 | nsyl 618 | . . 3 |
50 | 34, 49 | orim12i 749 | . 2 |
51 | 18, 50 | ax-mp 5 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 698 wceq 1342 wcel 2135 wne 2334 wral 2442 crab 2446 cvv 2721 c0 3404 csn 3570 cpr 3571 con0 4335 csuc 4337 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-nul 4102 ax-pow 4147 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2723 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-uni 3784 df-tr 4075 df-iord 4338 df-on 4340 df-suc 4343 |
This theorem is referenced by: ordsucunielexmid 4502 |
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