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Theorem enpr1g 8287
 Description: {𝐴, 𝐴} has only one element. (Contributed by FL, 15-Feb-2010.)
Assertion
Ref Expression
enpr1g (𝐴𝑉 → {𝐴, 𝐴} ≈ 1o)

Proof of Theorem enpr1g
StepHypRef Expression
1 dfsn2 4409 . 2 {𝐴} = {𝐴, 𝐴}
2 ensn1g 8286 . 2 (𝐴𝑉 → {𝐴} ≈ 1o)
31, 2syl5eqbrr 4908 1 (𝐴𝑉 → {𝐴, 𝐴} ≈ 1o)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2166  {csn 4396  {cpr 4398   class class class wbr 4872  1oc1o 7818   ≈ cen 8218 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2390  ax-ext 2802  ax-sep 5004  ax-nul 5012  ax-pr 5126  ax-un 7208 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2604  df-eu 2639  df-clab 2811  df-cleq 2817  df-clel 2820  df-nfc 2957  df-ral 3121  df-rex 3122  df-rab 3125  df-v 3415  df-dif 3800  df-un 3802  df-in 3804  df-ss 3811  df-nul 4144  df-if 4306  df-sn 4397  df-pr 4399  df-op 4403  df-uni 4658  df-br 4873  df-opab 4935  df-id 5249  df-xp 5347  df-rel 5348  df-cnv 5349  df-co 5350  df-dm 5351  df-rn 5352  df-suc 5968  df-fun 6124  df-fn 6125  df-f 6126  df-f1 6127  df-fo 6128  df-f1o 6129  df-1o 7825  df-en 8222 This theorem is referenced by:  pr2ne  9140
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