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Theorem reuen1 8998
Description: Two ways to express "exactly one". (Contributed by Stefan O'Rear, 28-Oct-2014.)
Assertion
Ref Expression
reuen1 (∃!𝑥𝐴 𝜑 ↔ {𝑥𝐴𝜑} ≈ 1o)
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem reuen1
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 reusn 4715 . 2 (∃!𝑥𝐴 𝜑 ↔ ∃𝑦{𝑥𝐴𝜑} = {𝑦})
2 en1 8994 . 2 ({𝑥𝐴𝜑} ≈ 1o ↔ ∃𝑦{𝑥𝐴𝜑} = {𝑦})
31, 2bitr4i 277 1 (∃!𝑥𝐴 𝜑 ↔ {𝑥𝐴𝜑} ≈ 1o)
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1541  wex 1781  ∃!wreu 3369  {crab 3425  {csn 4613   class class class wbr 5132  1oc1o 8432  cen 8909
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5283  ax-nul 5290  ax-pr 5411
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3372  df-rab 3426  df-v 3468  df-dif 3938  df-un 3940  df-in 3942  df-ss 3952  df-nul 4310  df-if 4514  df-sn 4614  df-pr 4616  df-op 4620  df-uni 4893  df-br 5133  df-opab 5195  df-id 5558  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-suc 6350  df-iota 6475  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-1o 8439  df-en 8913
This theorem is referenced by:  euen1  8999  isppw  26522
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