Theorem eqsn 4799

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} ↔ ∀𝑥 ∈ 𝐴 𝑥 = 𝐵))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem eqsn

Step	Hyp	Ref	Expression
1		df-ne 2965	. . 3 ⊢ (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅)
2		biorf 949	. . 3 ⊢ (¬ 𝐴 = ∅ → (𝐴 = {𝐵} ↔ (𝐴 = ∅ ∨ 𝐴 = {𝐵})))
3	1, 2	sylbi 220	. 2 ⊢ (𝐴 ≠ ∅ → (𝐴 = {𝐵} ↔ (𝐴 = ∅ ∨ 𝐴 = {𝐵})))
4		dfss3 3934	. . 3 ⊢ (𝐴 ⊆ {𝐵} ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ {𝐵})
5		sssn 4796	. . 3 ⊢ (𝐴 ⊆ {𝐵} ↔ (𝐴 = ∅ ∨ 𝐴 = {𝐵}))
6		velsn 4610	. . . 4 ⊢ (𝑥 ∈ {𝐵} ↔ 𝑥 = 𝐵)
7	6	ralbii 3117	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ∈ {𝐵} ↔ ∀𝑥 ∈ 𝐴 𝑥 = 𝐵)
8	4, 5, 7	3bitr3i 304	. 2 ⊢ ((𝐴 = ∅ ∨ 𝐴 = {𝐵}) ↔ ∀𝑥 ∈ 𝐴 𝑥 = 𝐵)
9	3, 8	bitrdi 290	1 ⊢ (𝐴 ≠ ∅ → (𝐴 = {𝐵} ↔ ∀𝑥 ∈ 𝐴 𝑥 = 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∨ wo 860   = wceq 1567   ∈ wcel 2149   ≠ wne 2964  ∀wral 3085   ⊆ wss 3913  ∅c0 4294  {csn 4594

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-ral 3086  df-v 3465  df-dif 3916  df-ss 3930  df-nul 4295  df-sn 4595

This theorem is referenced by:  eqsnd  4800  issn  4801  zornn0g  10489  hashgt12el  14459  hashgt12el2  14460  hashge2el2dif  14517  simpgnideld  20171  01eq0ring  20614  lssne0  21050  qtopeu  23842  n0nsnel  32802  dimval  33936  dimvalfi  33937  rngoueqz  38513  n0nsn2el  47685  lmod0rng  48917