Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  sosn Unicode version

Theorem sosn 5025
 Description: Strict ordering on a singleton. (Contributed by Mario Carneiro, 28-Dec-2014.)
Assertion
Ref Expression
sosn

Proof of Theorem sosn
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elsni 4018 . . . . . 6
2 elsni 4018 . . . . . . 7
32eqcomd 2462 . . . . . 6
41, 3sylan9eq 2515 . . . . 5
5 3mix2 1158 . . . . 5
64, 5syl 16 . . . 4
76rgen2a 2902 . . 3
8 df-so 4759 . . 3
97, 8mpbiran2 910 . 2
10 posn 5024 . 2
119, 10syl5bb 257 1
 Colors of variables: wff setvar class Syntax hints:  -.wn 3  ->wi 4  <->wb 184  /\wa 369  \/w3o 964  e.wcel 1758  A.wral 2800  {csn 3993   class class class wbr 4409  Powpo 4756  Orwor 4757  Relwrel 4962 This theorem is referenced by:  wesn  5027 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 1955  ax-ext 2432  ax-sep 4530  ax-nul 4538  ax-pr 4648 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-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2805  df-rex 2806  df-rab 2809  df-v 3083  df-sbc 3298  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3752  df-if 3906  df-sn 3994  df-pr 3996  df-op 4000  df-br 4410  df-opab 4468  df-po 4758  df-so 4759  df-xp 4963  df-rel 4964
 Copyright terms: Public domain W3C validator