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Theorem issn 4789
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 groups:   𝑥,𝐴,𝑦   𝑧,𝐴,𝑥

Proof of Theorem issn
StepHypRef Expression
1 equcom 2022 . . . . . 6 (𝑥 = 𝑦𝑦 = 𝑥)
21a1i 11 . . . . 5 (𝑥𝐴 → (𝑥 = 𝑦𝑦 = 𝑥))
32ralbidv 3173 . . . 4 (𝑥𝐴 → (∀𝑦𝐴 𝑥 = 𝑦 ↔ ∀𝑦𝐴 𝑦 = 𝑥))
4 ne0i 4293 . . . . 5 (𝑥𝐴𝐴 ≠ ∅)
5 eqsn 4788 . . . . 5 (𝐴 ≠ ∅ → (𝐴 = {𝑥} ↔ ∀𝑦𝐴 𝑦 = 𝑥))
64, 5syl 17 . . . 4 (𝑥𝐴 → (𝐴 = {𝑥} ↔ ∀𝑦𝐴 𝑦 = 𝑥))
73, 6bitr4d 282 . . 3 (𝑥𝐴 → (∀𝑦𝐴 𝑥 = 𝑦𝐴 = {𝑥}))
8 sneq 4595 . . . . 5 (𝑧 = 𝑥 → {𝑧} = {𝑥})
98eqeq2d 2749 . . . 4 (𝑧 = 𝑥 → (𝐴 = {𝑧} ↔ 𝐴 = {𝑥}))
109spcegv 3555 . . 3 (𝑥𝐴 → (𝐴 = {𝑥} → ∃𝑧 𝐴 = {𝑧}))
117, 10sylbid 239 . 2 (𝑥𝐴 → (∀𝑦𝐴 𝑥 = 𝑦 → ∃𝑧 𝐴 = {𝑧}))
1211rexlimiv 3144 1 (∃𝑥𝐴𝑦𝐴 𝑥 = 𝑦 → ∃𝑧 𝐴 = {𝑧})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1542  wex 1782  wcel 2107  wne 2942  wral 3063  wrex 3072  c0 4281  {csn 4585
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2943  df-ral 3064  df-rex 3073  df-v 3446  df-dif 3912  df-in 3916  df-ss 3926  df-nul 4282  df-sn 4586
This theorem is referenced by:  n0snor2el  4790  f1cdmsn  7225
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