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

Theorem enref 7874
 Description: Equinumerosity is reflexive. Theorem 1 of [Suppes] p. 92. (Contributed by NM, 25-Sep-2004.)
Hypothesis
Ref Expression
enref.1 𝐴 ∈ V
Assertion
Ref Expression
enref 𝐴𝐴

Proof of Theorem enref
StepHypRef Expression
1 enref.1 . 2 𝐴 ∈ V
2 enrefg 7873 . 2 (𝐴 ∈ V → 𝐴𝐴)
31, 2ax-mp 5 1 𝐴𝐴
 Colors of variables: wff setvar class Syntax hints:   ∈ wcel 1977  Vcvv 3173   class class class wbr 4583   ≈ cen 7838 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-en 7842 This theorem is referenced by:  ener  7888  enerOLD  7889  en0  7905  pwen  8018  phplem2  8025  phplem3  8026  isinf  8058  pssnn  8063  karden  8641  mappwen  8818  cdacomen  8886  infmap2  8923  ackbij1lem5  8929  axcc4dom  9146  domtriomlem  9147  cfpwsdom  9285  0tsk  9456  fzennn  12629  qnnen  14781  rpnnen  14795  rexpen  14796  lmisfree  20000  met2ndci  22137  lgseisenlem2  24901  poimirlem9  32588  poimirlem26  32605
 Copyright terms: Public domain W3C validator