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Theorem eqsn 4329
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 2791 . . 3 (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅)
2 biorf 420 . . 3 𝐴 = ∅ → (𝐴 = {𝐵} ↔ (𝐴 = ∅ ∨ 𝐴 = {𝐵})))
31, 2sylbi 207 . 2 (𝐴 ≠ ∅ → (𝐴 = {𝐵} ↔ (𝐴 = ∅ ∨ 𝐴 = {𝐵})))
4 dfss3 3573 . . 3 (𝐴 ⊆ {𝐵} ↔ ∀𝑥𝐴 𝑥 ∈ {𝐵})
5 sssn 4326 . . 3 (𝐴 ⊆ {𝐵} ↔ (𝐴 = ∅ ∨ 𝐴 = {𝐵}))
6 velsn 4164 . . . 4 (𝑥 ∈ {𝐵} ↔ 𝑥 = 𝐵)
76ralbii 2974 . . 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 383   = wceq 1480  wcel 1987  wne 2790  wral 2907  wss 3555  c0 3891  {csn 4148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-v 3188  df-dif 3558  df-in 3562  df-ss 3569  df-nul 3892  df-sn 4149
This theorem is referenced by:  issn  4331  zornn0g  9271  hashgt12el  13150  hashgt12el2  13151  hashge2el2dif  13200  lssne0  18870  qtopeu  21429  rngoueqz  33371  mapdm0  38857  lmod0rng  41156
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