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Theorem enpr1g 8174
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 4327 . 2 {𝐴} = {𝐴, 𝐴}
2 ensn1g 8173 . 2 (𝐴𝑉 → {𝐴} ≈ 1𝑜)
31, 2syl5eqbrr 4820 1 (𝐴𝑉 → {𝐴, 𝐴} ≈ 1𝑜)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2144  {csn 4314  {cpr 4316   class class class wbr 4784  1𝑜c1o 7705  cen 8105
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-suc 5872  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-1o 7712  df-en 8109
This theorem is referenced by:  pr2ne  9027
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