Theorem eqsn 4785

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 2933	. . 3 ⊢ (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅)
2		biorf 936	. . 3 ⊢ (¬ 𝐴 = ∅ → (𝐴 = {𝐵} ↔ (𝐴 = ∅ ∨ 𝐴 = {𝐵})))
3	1, 2	sylbi 217	. 2 ⊢ (𝐴 ≠ ∅ → (𝐴 = {𝐵} ↔ (𝐴 = ∅ ∨ 𝐴 = {𝐵})))
4		dfss3 3922	. . 3 ⊢ (𝐴 ⊆ {𝐵} ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ {𝐵})
5		sssn 4782	. . 3 ⊢ (𝐴 ⊆ {𝐵} ↔ (𝐴 = ∅ ∨ 𝐴 = {𝐵}))
6		velsn 4596	. . . 4 ⊢ (𝑥 ∈ {𝐵} ↔ 𝑥 = 𝐵)
7	6	ralbii 3082	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ∈ {𝐵} ↔ ∀𝑥 ∈ 𝐴 𝑥 = 𝐵)
8	4, 5, 7	3bitr3i 301	. 2 ⊢ ((𝐴 = ∅ ∨ 𝐴 = {𝐵}) ↔ ∀𝑥 ∈ 𝐴 𝑥 = 𝐵)
9	3, 8	bitrdi 287	1 ⊢ (𝐴 ≠ ∅ → (𝐴 = {𝐵} ↔ ∀𝑥 ∈ 𝐴 𝑥 = 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∨ wo 847   = wceq 1541   ∈ wcel 2113   ≠ wne 2932  ∀wral 3051   ⊆ wss 3901  ∅c0 4285  {csn 4580

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-v 3442  df-dif 3904  df-ss 3918  df-nul 4286  df-sn 4581

This theorem is referenced by:  eqsnd  4786  issn  4788  zornn0g  10415  hashgt12el  14345  hashgt12el2  14346  hashge2el2dif  14403  simpgnideld  20030  01eq0ring  20463  lssne0  20902  qtopeu  23660  n0nsnel  32590  dimval  33757  dimvalfi  33758  rngoueqz  38141  n0nsn2el  47281  lmod0rng  48485