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Theorem eqsn 4746
 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 3015 . . 3 (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅)
2 biorf 934 . . 3 𝐴 = ∅ → (𝐴 = {𝐵} ↔ (𝐴 = ∅ ∨ 𝐴 = {𝐵})))
31, 2sylbi 220 . 2 (𝐴 ≠ ∅ → (𝐴 = {𝐵} ↔ (𝐴 = ∅ ∨ 𝐴 = {𝐵})))
4 dfss3 3941 . . 3 (𝐴 ⊆ {𝐵} ↔ ∀𝑥𝐴 𝑥 ∈ {𝐵})
5 sssn 4743 . . 3 (𝐴 ⊆ {𝐵} ↔ (𝐴 = ∅ ∨ 𝐴 = {𝐵}))
6 velsn 4566 . . . 4 (𝑥 ∈ {𝐵} ↔ 𝑥 = 𝐵)
76ralbii 3160 . . 3 (∀𝑥𝐴 𝑥 ∈ {𝐵} ↔ ∀𝑥𝐴 𝑥 = 𝐵)
84, 5, 73bitr3i 304 . 2 ((𝐴 = ∅ ∨ 𝐴 = {𝐵}) ↔ ∀𝑥𝐴 𝑥 = 𝐵)
93, 8syl6bb 290 1 (𝐴 ≠ ∅ → (𝐴 = {𝐵} ↔ ∀𝑥𝐴 𝑥 = 𝐵))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∨ wo 844   = wceq 1538   ∈ wcel 2115   ≠ wne 3014  ∀wral 3133   ⊆ wss 3919  ∅c0 4276  {csn 4550 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-11 2162  ax-12 2179  ax-ext 2796 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-v 3482  df-dif 3922  df-in 3926  df-ss 3936  df-nul 4277  df-sn 4551 This theorem is referenced by:  issn  4747  zornn0g  9925  hashgt12el  13788  hashgt12el2  13789  hashge2el2dif  13843  simpgnideld  19221  lssne0  19722  qtopeu  22324  dimval  31061  dimvalfi  31062  rngoueqz  35323  lmod0rng  44418
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