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

Theorem enref 6890
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 6889 . 2  |-  ( A  e.  _V  ->  A  ~~  A )
31, 2ax-mp 10 1  |-  A  ~~  A
Colors of variables: wff set class
Syntax hints:    e. wcel 1685   _Vcvv 2790   class class class wbr 4025    ~~ cen 6856
This theorem is referenced by:  ener  6904  en0  6920  pwen  7030  phplem2  7037  phplem3  7038  isinf  7072  pssnn  7077  karden  7561  mappwen  7735  cdacomen  7803  infmap2  7840  ackbij1lem5  7846  axcc4dom  8063  domtriomlem  8064  cfpwsdom  8202  0tsk  8373  fzennn  11025  qnnen  12487  rpnnen  12500  rexpen  12501  met2ndci  18063  lgseisenlem2  20584  lmisfree  26712
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-br 4026  df-opab 4080  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-fun 5224  df-fn 5225  df-f 5226  df-f1 5227  df-fo 5228  df-f1o 5229  df-en 6860
  Copyright terms: Public domain W3C validator