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

Theorem enref 7131
Description: Equinumerosity is reflexive. Theorem 1 of [Suppes] p. 92. (Contributed by NM, 25-Sep-2004.)
Hypothesis
Ref Expression
enref.1  |-  A  e. 
_V
Assertion
Ref Expression
enref  |-  A  ~~  A

Proof of Theorem enref
StepHypRef Expression
1 enref.1 . 2  |-  A  e. 
_V
2 enrefg 7130 . 2  |-  ( A  e.  _V  ->  A  ~~  A )
31, 2ax-mp 8 1  |-  A  ~~  A
Colors of variables: wff set class
Syntax hints:    e. wcel 1725   _Vcvv 2948   class class class wbr 4204    ~~ cen 7097
This theorem is referenced by:  ener  7145  en0  7161  pwen  7271  phplem2  7278  phplem3  7279  isinf  7313  pssnn  7318  karden  7808  mappwen  7982  cdacomen  8050  infmap2  8087  ackbij1lem5  8093  axcc4dom  8310  domtriomlem  8311  cfpwsdom  8448  0tsk  8619  fzennn  11295  qnnen  12801  rpnnen  12814  rexpen  12815  met2ndci  18540  lgseisenlem2  21122  lmisfree  27227
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-en 7101
  Copyright terms: Public domain W3C validator