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Theorem issn 4726
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 2025 . . . . . 6 (𝑥 = 𝑦𝑦 = 𝑥)
21a1i 11 . . . . 5 (𝑥𝐴 → (𝑥 = 𝑦𝑦 = 𝑥))
32ralbidv 3165 . . . 4 (𝑥𝐴 → (∀𝑦𝐴 𝑥 = 𝑦 ↔ ∀𝑦𝐴 𝑦 = 𝑥))
4 ne0i 4253 . . . . 5 (𝑥𝐴𝐴 ≠ ∅)
5 eqsn 4725 . . . . 5 (𝐴 ≠ ∅ → (𝐴 = {𝑥} ↔ ∀𝑦𝐴 𝑦 = 𝑥))
64, 5syl 17 . . . 4 (𝑥𝐴 → (𝐴 = {𝑥} ↔ ∀𝑦𝐴 𝑦 = 𝑥))
73, 6bitr4d 285 . . 3 (𝑥𝐴 → (∀𝑦𝐴 𝑥 = 𝑦𝐴 = {𝑥}))
8 sneq 4538 . . . . 5 (𝑧 = 𝑥 → {𝑧} = {𝑥})
98eqeq2d 2812 . . . 4 (𝑧 = 𝑥 → (𝐴 = {𝑧} ↔ 𝐴 = {𝑥}))
109spcegv 3548 . . 3 (𝑥𝐴 → (𝐴 = {𝑥} → ∃𝑧 𝐴 = {𝑧}))
117, 10sylbid 243 . 2 (𝑥𝐴 → (∀𝑦𝐴 𝑥 = 𝑦 → ∃𝑧 𝐴 = {𝑧}))
1211rexlimiv 3242 1 (∃𝑥𝐴𝑦𝐴 𝑥 = 𝑦 → ∃𝑧 𝐴 = {𝑧})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209   = wceq 1538  wex 1781  wcel 2112  wne 2990  wral 3109  wrex 3110  c0 4246  {csn 4528
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-12 2176  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-ne 2991  df-ral 3114  df-rex 3115  df-v 3446  df-dif 3887  df-in 3891  df-ss 3901  df-nul 4247  df-sn 4529
This theorem is referenced by:  n0snor2el  4727
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