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Theorem eqsn 4789
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 2944 . . 3 (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅)
2 biorf 935 . . 3 𝐴 = ∅ → (𝐴 = {𝐵} ↔ (𝐴 = ∅ ∨ 𝐴 = {𝐵})))
31, 2sylbi 216 . 2 (𝐴 ≠ ∅ → (𝐴 = {𝐵} ↔ (𝐴 = ∅ ∨ 𝐴 = {𝐵})))
4 dfss3 3932 . . 3 (𝐴 ⊆ {𝐵} ↔ ∀𝑥𝐴 𝑥 ∈ {𝐵})
5 sssn 4786 . . 3 (𝐴 ⊆ {𝐵} ↔ (𝐴 = ∅ ∨ 𝐴 = {𝐵}))
6 velsn 4602 . . . 4 (𝑥 ∈ {𝐵} ↔ 𝑥 = 𝐵)
76ralbii 3096 . . 3 (∀𝑥𝐴 𝑥 ∈ {𝐵} ↔ ∀𝑥𝐴 𝑥 = 𝐵)
84, 5, 73bitr3i 300 . 2 ((𝐴 = ∅ ∨ 𝐴 = {𝐵}) ↔ ∀𝑥𝐴 𝑥 = 𝐵)
93, 8bitrdi 286 1 (𝐴 ≠ ∅ → (𝐴 = {𝐵} ↔ ∀𝑥𝐴 𝑥 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wo 845   = wceq 1541  wcel 2106  wne 2943  wral 3064  wss 3910  c0 4282  {csn 4586
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2944  df-ral 3065  df-v 3447  df-dif 3913  df-in 3917  df-ss 3927  df-nul 4283  df-sn 4587
This theorem is referenced by:  issn  4790  zornn0g  10440  hashgt12el  14321  hashgt12el2  14322  hashge2el2dif  14378  simpgnideld  19876  lssne0  20409  qtopeu  23065  dimval  32291  dimvalfi  32292  rngoueqz  36390  lmod0rng  46138
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