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Theorem euen1 7912
Description: Two ways to express "exactly one". (Contributed by Stefan O'Rear, 28-Oct-2014.)
Assertion
Ref Expression
euen1 (∃!𝑥𝜑 ↔ {𝑥𝜑} ≈ 1𝑜)

Proof of Theorem euen1
StepHypRef Expression
1 reuen1 7911 . 2 (∃!𝑥 ∈ V 𝜑 ↔ {𝑥 ∈ V ∣ 𝜑} ≈ 1𝑜)
2 reuv 3194 . 2 (∃!𝑥 ∈ V 𝜑 ↔ ∃!𝑥𝜑)
3 rabab 3196 . . 3 {𝑥 ∈ V ∣ 𝜑} = {𝑥𝜑}
43breq1i 4590 . 2 ({𝑥 ∈ V ∣ 𝜑} ≈ 1𝑜 ↔ {𝑥𝜑} ≈ 1𝑜)
51, 2, 43bitr3i 289 1 (∃!𝑥𝜑 ↔ {𝑥𝜑} ≈ 1𝑜)
Colors of variables: wff setvar class
Syntax hints:  wb 195  ∃!weu 2458  {cab 2596  ∃!wreu 2898  {crab 2900  Vcvv 3173   class class class wbr 4583  1𝑜c1o 7440  cen 7838
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-1o 7447  df-en 7842
This theorem is referenced by:  euen1b  7913  modom  8046
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