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Theorem eqsn 4301
 Description: Two ways to express that a nonempty set equals a singleton. (Contributed by NM, 15-Dec-2007.) (Proof shortened by JJ, 23-Jul-2021.)
Assertion
Ref Expression
eqsn (𝐴 ≠ ∅ → (𝐴 = {𝐵} ↔ ∀𝑥𝐴 𝑥 = 𝐵))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem eqsn
StepHypRef Expression
1 df-ne 2782 . . 3 (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅)
2 biorf 419 . . 3 𝐴 = ∅ → (𝐴 = {𝐵} ↔ (𝐴 = ∅ ∨ 𝐴 = {𝐵})))
31, 2sylbi 206 . 2 (𝐴 ≠ ∅ → (𝐴 = {𝐵} ↔ (𝐴 = ∅ ∨ 𝐴 = {𝐵})))
4 dfss3 3558 . . 3 (𝐴 ⊆ {𝐵} ↔ ∀𝑥𝐴 𝑥 ∈ {𝐵})
5 sssn 4298 . . 3 (𝐴 ⊆ {𝐵} ↔ (𝐴 = ∅ ∨ 𝐴 = {𝐵}))
6 velsn 4141 . . . 4 (𝑥 ∈ {𝐵} ↔ 𝑥 = 𝐵)
76ralbii 2963 . . 3 (∀𝑥𝐴 𝑥 ∈ {𝐵} ↔ ∀𝑥𝐴 𝑥 = 𝐵)
84, 5, 73bitr3i 289 . 2 ((𝐴 = ∅ ∨ 𝐴 = {𝐵}) ↔ ∀𝑥𝐴 𝑥 = 𝐵)
93, 8syl6bb 275 1 (𝐴 ≠ ∅ → (𝐴 = {𝐵} ↔ ∀𝑥𝐴 𝑥 = 𝐵))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896   ⊆ wss 3540  ∅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-v 3175  df-dif 3543  df-in 3547  df-ss 3554  df-nul 3875  df-sn 4126 This theorem is referenced by:  issn  4303  zornn0g  9210  hashgt12el  13070  hashgt12el2  13071  hashge2el2dif  13117  lssne0  18772  qtopeu  21329  rngoueqz  32909  mapdm0  38378  lmod0rng  41658
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