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

Theorem sniota 5517
 Description: A class abstraction with a unique member can be expressed as a singleton. (Contributed by Mario Carneiro, 23-Dec-2016.)
Assertion
Ref Expression
sniota

Proof of Theorem sniota
StepHypRef Expression
1 nfeu1 2275 . . 3
2 iota1 5504 . . . . 5
3 eqcom 2463 . . . . 5
42, 3syl6bb 261 . . . 4
5 abid 2441 . . . 4
6 vex 3081 . . . . 5
76elsnc 4010 . . . 4
84, 5, 73bitr4g 288 . . 3
91, 8alrimi 1816 . 2
10 nfab1 2618 . . 3
11 nfiota1 5492 . . . 4
1211nfsn 4043 . . 3
1310, 12cleqf 2643 . 2
149, 13sylibr 212 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184  wal 1368   wceq 1370   wcel 1758  weu 2262  cab 2439  csn 3986  cio 5488 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-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2804  df-rex 2805  df-v 3080  df-sbc 3295  df-un 3442  df-sn 3987  df-pr 3989  df-uni 4201  df-iota 5490 This theorem is referenced by:  snriota  6192
 Copyright terms: Public domain W3C validator