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Theorem enref 7538
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 7537 . 2  |-  ( A  e.  _V  ->  A  ~~  A )
31, 2ax-mp 5 1  |-  A  ~~  A
Colors of variables: wff setvar class
Syntax hints:    e. wcel 1762   _Vcvv 3106   class class class wbr 4440    ~~ cen 7503
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-en 7507
This theorem is referenced by:  ener  7552  en0  7568  pwen  7680  phplem2  7687  phplem3  7688  isinf  7723  pssnn  7728  karden  8302  mappwen  8482  cdacomen  8550  infmap2  8587  ackbij1lem5  8593  axcc4dom  8810  domtriomlem  8811  cfpwsdom  8948  0tsk  9122  fzennn  12034  qnnen  13797  rpnnen  13810  rexpen  13811  lmisfree  18637  met2ndci  20753  lgseisenlem2  23346
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