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Theorem eqsn 4498
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 2925 . . 3 (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅)
2 biorf 419 . . 3 𝐴 = ∅ → (𝐴 = {𝐵} ↔ (𝐴 = ∅ ∨ 𝐴 = {𝐵})))
31, 2sylbi 207 . 2 (𝐴 ≠ ∅ → (𝐴 = {𝐵} ↔ (𝐴 = ∅ ∨ 𝐴 = {𝐵})))
4 dfss3 3725 . . 3 (𝐴 ⊆ {𝐵} ↔ ∀𝑥𝐴 𝑥 ∈ {𝐵})
5 sssn 4495 . . 3 (𝐴 ⊆ {𝐵} ↔ (𝐴 = ∅ ∨ 𝐴 = {𝐵}))
6 velsn 4329 . . . 4 (𝑥 ∈ {𝐵} ↔ 𝑥 = 𝐵)
76ralbii 3110 . . 3 (∀𝑥𝐴 𝑥 ∈ {𝐵} ↔ ∀𝑥𝐴 𝑥 = 𝐵)
84, 5, 73bitr3i 290 . 2 ((𝐴 = ∅ ∨ 𝐴 = {𝐵}) ↔ ∀𝑥𝐴 𝑥 = 𝐵)
93, 8syl6bb 276 1 (𝐴 ≠ ∅ → (𝐴 = {𝐵} ↔ ∀𝑥𝐴 𝑥 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382   = wceq 1624  wcel 2131  wne 2924  wral 3042  wss 3707  c0 4050  {csn 4313
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-v 3334  df-dif 3710  df-in 3714  df-ss 3721  df-nul 4051  df-sn 4314
This theorem is referenced by:  issn  4500  zornn0g  9511  hashgt12el  13394  hashgt12el2  13395  hashge2el2dif  13446  lssne0  19145  qtopeu  21713  rngoueqz  34044  mapdm0OLD  39874  lmod0rng  42370
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