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Theorem enref 7169
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 7168 . 2  |-  ( A  e.  _V  ->  A  ~~  A )
31, 2ax-mp 5 1  |-  A  ~~  A
Colors of variables: wff set class
Syntax hints:    e. wcel 1727   _Vcvv 2962   class class class wbr 4237    ~~ cen 7135
This theorem is referenced by:  ener  7183  en0  7199  pwen  7309  phplem2  7316  phplem3  7317  isinf  7351  pssnn  7356  karden  7850  mappwen  8024  cdacomen  8092  infmap2  8129  ackbij1lem5  8135  axcc4dom  8352  domtriomlem  8353  cfpwsdom  8490  0tsk  8661  fzennn  11338  qnnen  12844  rpnnen  12857  rexpen  12858  met2ndci  18583  lgseisenlem2  21165  lmisfree  27327
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-rab 2720  df-v 2964  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-br 4238  df-opab 4292  df-id 4527  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-en 7139
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