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Mirrors > Home > ILE Home > Th. List > onntri35 | Unicode version |
Description: Double negated ordinal
trichotomy.
There are five equivalent statements: (1) , (2) , (3) , (4) , and (5) EXMID. That these are all equivalent is expressed by (1) implies (3) (onntri13 7167), (3) implies (5) (onntri35 7166), (5) implies (1) (onntri51 7169), (2) implies (4) (onntri24 7171), (4) implies (5) (onntri45 7170), and (5) implies (2) (onntri52 7173). Another way of stating this is that EXMID is equivalent to trichotomy, either the or the form, as shown in exmidontri 7168 and exmidontri2or 7172, respectively. Thus EXMID is equivalent to (1) or (2). In addition, EXMID is equivalent to (3) by onntri3or 7174 and (4) by onntri2or 7175. (Contributed by James E. Hanson and Jim Kingdon, 2-Aug-2024.) |
Ref | Expression |
---|---|
onntri35 | EXMID |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pw1on 7155 | . . . . 5 | |
2 | 1 | onsuci 4474 | . . . 4 |
3 | 3on 6371 | . . . 4 | |
4 | eleq1 2220 | . . . . . . . 8 | |
5 | eqeq1 2164 | . . . . . . . 8 | |
6 | eleq2 2221 | . . . . . . . 8 | |
7 | 4, 5, 6 | 3orbi123d 1293 | . . . . . . 7 |
8 | 7 | notbid 657 | . . . . . 6 |
9 | 8 | notbid 657 | . . . . 5 |
10 | eleq2 2221 | . . . . . . . 8 | |
11 | eqeq2 2167 | . . . . . . . 8 | |
12 | eleq1 2220 | . . . . . . . 8 | |
13 | 10, 11, 12 | 3orbi123d 1293 | . . . . . . 7 |
14 | 13 | notbid 657 | . . . . . 6 |
15 | 14 | notbid 657 | . . . . 5 |
16 | 9, 15 | rspc2v 2829 | . . . 4 |
17 | 2, 3, 16 | mp2an 423 | . . 3 |
18 | 3ioran 978 | . . 3 | |
19 | 17, 18 | sylnib 666 | . 2 |
20 | sucpw1nel3 7162 | . . . 4 | |
21 | 20 | a1i 9 | . . 3 EXMID |
22 | 2on 6369 | . . . . . . 7 | |
23 | suc11 4516 | . . . . . . 7 | |
24 | 1, 22, 23 | mp2an 423 | . . . . . 6 |
25 | df-3o 6362 | . . . . . . 7 | |
26 | 25 | eqeq2i 2168 | . . . . . 6 |
27 | exmidpweq 6851 | . . . . . 6 EXMID | |
28 | 24, 26, 27 | 3bitr4ri 212 | . . . . 5 EXMID |
29 | 28 | notbii 658 | . . . 4 EXMID |
30 | 29 | biimpi 119 | . . 3 EXMID |
31 | 3nelsucpw1 7163 | . . . 4 | |
32 | 31 | a1i 9 | . . 3 EXMID |
33 | 21, 30, 32 | 3jca 1162 | . 2 EXMID |
34 | 19, 33 | nsyl 618 | 1 EXMID |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wb 104 w3o 962 w3a 963 wceq 1335 wcel 2128 wral 2435 cpw 3543 EXMIDwem 4155 con0 4323 csuc 4325 c1o 6353 c2o 6354 c3o 6355 |
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 4082 ax-nul 4090 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4495 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 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-v 2714 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-pw 3545 df-sn 3566 df-pr 3567 df-uni 3773 df-int 3808 df-tr 4063 df-exmid 4156 df-iord 4326 df-on 4328 df-suc 4331 df-iom 4549 df-1o 6360 df-2o 6361 df-3o 6362 |
This theorem is referenced by: onntri3or 7174 |
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