Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  eqsnm GIF version

Theorem eqsnm 3682
 Description: Two ways to express that an inhabited set equals a singleton. (Contributed by Jim Kingdon, 11-Aug-2018.)
Assertion
Ref Expression
eqsnm (∃𝑥 𝑥𝐴 → (𝐴 = {𝐵} ↔ ∀𝑥𝐴 𝑥 = 𝐵))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem eqsnm
StepHypRef Expression
1 sssnm 3681 . 2 (∃𝑥 𝑥𝐴 → (𝐴 ⊆ {𝐵} ↔ 𝐴 = {𝐵}))
2 dfss3 3087 . . 3 (𝐴 ⊆ {𝐵} ↔ ∀𝑥𝐴 𝑥 ∈ {𝐵})
3 velsn 3544 . . . 4 (𝑥 ∈ {𝐵} ↔ 𝑥 = 𝐵)
43ralbii 2441 . . 3 (∀𝑥𝐴 𝑥 ∈ {𝐵} ↔ ∀𝑥𝐴 𝑥 = 𝐵)
52, 4bitri 183 . 2 (𝐴 ⊆ {𝐵} ↔ ∀𝑥𝐴 𝑥 = 𝐵)
61, 5bitr3di 194 1 (∃𝑥 𝑥𝐴 → (𝐴 = {𝐵} ↔ ∀𝑥𝐴 𝑥 = 𝐵))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 104   = wceq 1331  ∃wex 1468   ∈ wcel 1480  ∀wral 2416   ⊆ wss 3071  {csn 3527 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-v 2688  df-in 3077  df-ss 3084  df-sn 3533 This theorem is referenced by:  nninfall  13234
 Copyright terms: Public domain W3C validator