Theorem eqsn 4797

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 2961	. . 3 ⊢ (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅)
2		biorf 950	. . 3 ⊢ (¬ 𝐴 = ∅ → (𝐴 = {𝐵} ↔ (𝐴 = ∅ ∨ 𝐴 = {𝐵})))
3	1, 2	sylbi 220	. 2 ⊢ (𝐴 ≠ ∅ → (𝐴 = {𝐵} ↔ (𝐴 = ∅ ∨ 𝐴 = {𝐵})))
4		dfss3 3927	. . 3 ⊢ (𝐴 ⊆ {𝐵} ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ {𝐵})
5		sssn 4794	. . 3 ⊢ (𝐴 ⊆ {𝐵} ↔ (𝐴 = ∅ ∨ 𝐴 = {𝐵}))
6		velsn 4607	. . . 4 ⊢ (𝑥 ∈ {𝐵} ↔ 𝑥 = 𝐵)
7	6	ralbii 3113	. . 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))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∨ wo 861   = wceq 1570   ∈ wcel 2146   ≠ wne 2960  ∀wral 3081   ⊆ wss 3906  ∅c0 4286  {csn 4591

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3082  df-v 3459  df-dif 3909  df-ss 3923  df-nul 4287  df-sn 4592

This theorem is used by:  eqsnd  4798  issn  4799  zornn0g  10504  hashgt12el  14479  hashgt12el2  14480  hashge2el2dif  14537  simpgnideld  20217  01eq0ring  20680  lssne0  21124  qtopeu  23926  n0nsnel  32934  dimval  34057  dimvalfi  34058  rngoueqz  38651  n0nsn2el  47822  lmod0rng  49053