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Theorem onntri35 7155
 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 7156), (3) implies (5) (onntri35 7155), (5) implies (1) (onntri51 7158), (2) implies (4) (onntri24 7160), (4) implies (5) (onntri45 7159), and (5) implies (2) (onntri52 7162). Another way of stating this is that EXMID is equivalent to trichotomy, either the or the form, as shown in exmidontri 7157 and exmidontri2or 7161, respectively. Thus EXMID is equivalent to (1) or (2). In addition, EXMID is equivalent to (3) by onntri3or 7163 and (4) by onntri2or 7164. (Contributed by James E. Hanson and Jim Kingdon, 2-Aug-2024.)
Assertion
Ref Expression
onntri35 EXMID
Distinct variable group:   ,

Proof of Theorem onntri35
StepHypRef Expression
1 pw1on 7144 . . . . 5
21onsuci 4473 . . . 4
3 3on 6368 . . . 4
4 eleq1 2220 . . . . . . . 8
5 eqeq1 2164 . . . . . . . 8
6 eleq2 2221 . . . . . . . 8
74, 5, 63orbi123d 1293 . . . . . . 7
87notbid 657 . . . . . 6
98notbid 657 . . . . 5
10 eleq2 2221 . . . . . . . 8
11 eqeq2 2167 . . . . . . . 8
12 eleq1 2220 . . . . . . . 8
1310, 11, 123orbi123d 1293 . . . . . . 7
1413notbid 657 . . . . . 6
1514notbid 657 . . . . 5
169, 15rspc2v 2829 . . . 4
172, 3, 16mp2an 423 . . 3
18 3ioran 978 . . 3
1917, 18sylnib 666 . 2
20 sucpw1nel3 7151 . . . 4
2120a1i 9 . . 3 EXMID
22 2on 6366 . . . . . . 7
23 suc11 4515 . . . . . . 7
241, 22, 23mp2an 423 . . . . . 6
25 df-3o 6359 . . . . . . 7
2625eqeq2i 2168 . . . . . 6
27 exmidpweq 6847 . . . . . 6 EXMID
2824, 26, 273bitr4ri 212 . . . . 5 EXMID
2928notbii 658 . . . 4 EXMID
3029biimpi 119 . . 3 EXMID
31 3nelsucpw1 7152 . . . 4
3231a1i 9 . . 3 EXMID
3321, 30, 323jca 1162 . 2 EXMID
3419, 33nsyl 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 4154  con0 4322   csuc 4324  c1o 6350  c2o 6351  c3o 6352 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 4134  ax-pr 4168  ax-un 4392  ax-setind 4494 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 4155  df-iord 4325  df-on 4327  df-suc 4330  df-iom 4548  df-1o 6357  df-2o 6358  df-3o 6359 This theorem is referenced by:  onntri3or  7163
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