Mathbox for Scott Fenton < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  socnv Structured version   Unicode version

Theorem socnv 27712
 Description: The converse of a strict ordering is still a strict ordering. (Contributed by Scott Fenton, 13-Jun-2018.)
Assertion
Ref Expression
socnv

Proof of Theorem socnv
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sopo 4759 . . 3
2 pocnv 27711 . . 3
31, 2syl 16 . 2
4 solin 4765 . . 3
5 vex 3074 . . . . . 6
6 vex 3074 . . . . . 6
75, 6brcnv 5123 . . . . 5
8 biid 236 . . . . 5
96, 5brcnv 5123 . . . . 5
107, 8, 93orbi123i 1178 . . . 4
11 orcom 387 . . . . . 6
1211orbi2i 519 . . . . 5
13 3orass 968 . . . . . 6
14 or12 523 . . . . . 6
1513, 14bitri 249 . . . . 5
16 3orass 968 . . . . . 6
17 or12 523 . . . . . 6
1816, 17bitri 249 . . . . 5
1912, 15, 183bitr4i 277 . . . 4
2010, 19bitri 249 . . 3
214, 20sylibr 212 . 2
223, 21issod 4772 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wo 368   wa 369   w3o 964   wcel 1758   class class class wbr 4393   wpo 4740   wor 4741  ccnv 4940 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pr 4632 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rab 2804  df-v 3073  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-sn 3979  df-pr 3981  df-op 3985  df-br 4394  df-opab 4452  df-po 4742  df-so 4743  df-cnv 4949 This theorem is referenced by:  wzel  27898  wsucex  27900  wsuccl  27901  wsuclb  27902
 Copyright terms: Public domain W3C validator