Theorem reuen1 8965

Description: Two ways to express "exactly one". (Contributed by Stefan O'Rear, 28-Oct-2014.)

Assertion

Ref	Expression
reuen1	⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ {𝑥 ∈ 𝐴 ∣ 𝜑} ≈ 1o)

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem reuen1

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		reusn 4686	. 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦{𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑦})
2		en1 8961	. 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ≈ 1o ↔ ∃𝑦{𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑦})
3	1, 2	bitr4i 277	1 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ {𝑥 ∈ 𝐴 ∣ 𝜑} ≈ 1o)

Syntax hints:   ↔ wb 205   = wceq 1541  ∃wex 1781  ∃!wreu 3349  {crab 3405  {csn 4584   class class class wbr 5103  1_oc1o 8401   ≈ cen 8876

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 2707  ax-sep 5254  ax-nul 5261  ax-pr 5382

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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3352  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-br 5104  df-opab 5166  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-suc 6321  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-1o 8408  df-en 8880

This theorem is referenced by:  euen1  8966  isppw  26447