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

Theorem issn 4303
 Description: A sufficient condition for a (nonempty) set to be a singleton. (Contributed by AV, 20-Sep-2020.)
Assertion
Ref Expression
issn (∃𝑥𝐴𝑦𝐴 𝑥 = 𝑦 → ∃𝑧 𝐴 = {𝑧})
Distinct variable group:   𝑥,𝐴,𝑦,𝑧

Proof of Theorem issn
StepHypRef Expression
1 equcom 1932 . . . . . . 7 (𝑥 = 𝑦𝑦 = 𝑥)
21a1i 11 . . . . . 6 (𝑥𝐴 → (𝑥 = 𝑦𝑦 = 𝑥))
32ralbidv 2969 . . . . 5 (𝑥𝐴 → (∀𝑦𝐴 𝑥 = 𝑦 ↔ ∀𝑦𝐴 𝑦 = 𝑥))
4 ne0i 3880 . . . . . 6 (𝑥𝐴𝐴 ≠ ∅)
5 eqsn 4301 . . . . . 6 (𝐴 ≠ ∅ → (𝐴 = {𝑥} ↔ ∀𝑦𝐴 𝑦 = 𝑥))
64, 5syl 17 . . . . 5 (𝑥𝐴 → (𝐴 = {𝑥} ↔ ∀𝑦𝐴 𝑦 = 𝑥))
73, 6bitr4d 270 . . . 4 (𝑥𝐴 → (∀𝑦𝐴 𝑥 = 𝑦𝐴 = {𝑥}))
8 sneq 4135 . . . . . 6 (𝑧 = 𝑥 → {𝑧} = {𝑥})
98eqeq2d 2620 . . . . 5 (𝑧 = 𝑥 → (𝐴 = {𝑧} ↔ 𝐴 = {𝑥}))
109spcegv 3267 . . . 4 (𝑥𝐴 → (𝐴 = {𝑥} → ∃𝑧 𝐴 = {𝑧}))
117, 10sylbid 229 . . 3 (𝑥𝐴 → (∀𝑦𝐴 𝑥 = 𝑦 → ∃𝑧 𝐴 = {𝑧}))
1211imp 444 . 2 ((𝑥𝐴 ∧ ∀𝑦𝐴 𝑥 = 𝑦) → ∃𝑧 𝐴 = {𝑧})
1312rexlimiva 3010 1 (∃𝑥𝐴𝑦𝐴 𝑥 = 𝑦 → ∃𝑧 𝐴 = {𝑧})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475  ∃wex 1695   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  ∅c0 3874  {csn 4125 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-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 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-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-v 3175  df-dif 3543  df-in 3547  df-ss 3554  df-nul 3875  df-sn 4126 This theorem is referenced by:  n0snor2el  4304
 Copyright terms: Public domain W3C validator