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

Theorem enref 8573
 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 8572 . 2 (𝐴 ∈ V → 𝐴𝐴)
31, 2ax-mp 5 1 𝐴𝐴
 Colors of variables: wff setvar class Syntax hints:   ∈ wcel 2111  Vcvv 3409   class class class wbr 5036   ≈ cen 8537 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302  ax-un 7465 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5037  df-opab 5099  df-id 5434  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-en 8541 This theorem is referenced by:  ener  8587  en0OLD  8604  pwen  8725  phplem2  8732  phplem3  8733  isinf  8782  pssnnOLD  8787  karden  9370  mappwen  9585  nnadju  9670  infmap2  9691  ackbij1lem5  9697  axcc4dom  9914  domtriomlem  9915  cfpwsdom  10057  0tsk  10228  fzennn  13398  qnnen  15627  rpnnen  15641  rexpen  15642  lmisfree  20620  met2ndci  23237  lgseisenlem2  26072  poimirlem9  35380  poimirlem26  35397  1aryenef  45473  2aryenef  45484
 Copyright terms: Public domain W3C validator