MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  snriota Structured version   Visualization version   GIF version

Theorem snriota 6833
Description: A restricted class abstraction with a unique member can be expressed as a singleton. (Contributed by NM, 30-May-2006.)
Assertion
Ref Expression
snriota (∃!𝑥𝐴 𝜑 → {𝑥𝐴𝜑} = {(𝑥𝐴 𝜑)})

Proof of Theorem snriota
StepHypRef Expression
1 df-reu 3062 . . 3 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥(𝑥𝐴𝜑))
2 sniota 6058 . . 3 (∃!𝑥(𝑥𝐴𝜑) → {𝑥 ∣ (𝑥𝐴𝜑)} = {(℩𝑥(𝑥𝐴𝜑))})
31, 2sylbi 208 . 2 (∃!𝑥𝐴 𝜑 → {𝑥 ∣ (𝑥𝐴𝜑)} = {(℩𝑥(𝑥𝐴𝜑))})
4 df-rab 3064 . 2 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
5 df-riota 6803 . . 3 (𝑥𝐴 𝜑) = (℩𝑥(𝑥𝐴𝜑))
65sneqi 4345 . 2 {(𝑥𝐴 𝜑)} = {(℩𝑥(𝑥𝐴𝜑))}
73, 4, 63eqtr4g 2824 1 (∃!𝑥𝐴 𝜑 → {𝑥𝐴𝜑} = {(𝑥𝐴 𝜑)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1652  wcel 2155  ∃!weu 2581  {cab 2751  ∃!wreu 3057  {crab 3059  {csn 4334  cio 6029  crio 6802
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-un 3737  df-sn 4335  df-pr 4337  df-uni 4595  df-iota 6031  df-riota 6803
This theorem is referenced by:  divalgmod  15411
  Copyright terms: Public domain W3C validator