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

Theorem enref 7973
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 7972 . 2 (𝐴 ∈ V → 𝐴𝐴)
31, 2ax-mp 5 1 𝐴𝐴
Colors of variables: wff setvar class
Syntax hints:  wcel 1988  Vcvv 3195   class class class wbr 4644  cen 7937
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-en 7941
This theorem is referenced by:  ener  7987  enerOLD  7988  en0  8004  pwen  8118  phplem2  8125  phplem3  8126  isinf  8158  pssnn  8163  karden  8743  mappwen  8920  cdacomen  8988  infmap2  9025  ackbij1lem5  9031  axcc4dom  9248  domtriomlem  9249  cfpwsdom  9391  0tsk  9562  fzennn  12750  qnnen  14923  rpnnen  14937  rexpen  14938  lmisfree  20162  met2ndci  22308  lgseisenlem2  25082  poimirlem9  33389  poimirlem26  33406
  Copyright terms: Public domain W3C validator