Theorem eqsn 4301

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 2782	. . 3 ⊢ (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅)
2		biorf 419	. . 3 ⊢ (¬ 𝐴 = ∅ → (𝐴 = {𝐵} ↔ (𝐴 = ∅ ∨ 𝐴 = {𝐵})))
3	1, 2	sylbi 206	. 2 ⊢ (𝐴 ≠ ∅ → (𝐴 = {𝐵} ↔ (𝐴 = ∅ ∨ 𝐴 = {𝐵})))
4		dfss3 3558	. . 3 ⊢ (𝐴 ⊆ {𝐵} ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ {𝐵})
5		sssn 4298	. . 3 ⊢ (𝐴 ⊆ {𝐵} ↔ (𝐴 = ∅ ∨ 𝐴 = {𝐵}))
6		velsn 4141	. . . 4 ⊢ (𝑥 ∈ {𝐵} ↔ 𝑥 = 𝐵)
7	6	ralbii 2963	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ∈ {𝐵} ↔ ∀𝑥 ∈ 𝐴 𝑥 = 𝐵)
8	4, 5, 7	3bitr3i 289	. 2 ⊢ ((𝐴 = ∅ ∨ 𝐴 = {𝐵}) ↔ ∀𝑥 ∈ 𝐴 𝑥 = 𝐵)
9	3, 8	syl6bb 275	1 ⊢ (𝐴 ≠ ∅ → (𝐴 = {𝐵} ↔ ∀𝑥 ∈ 𝐴 𝑥 = 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896   ⊆ wss 3540  ∅c0 3874  {csn 4125

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-v 3175  df-dif 3543  df-in 3547  df-ss 3554  df-nul 3875  df-sn 4126

This theorem is referenced by:  issn  4303  zornn0g  9210  hashgt12el  13070  hashgt12el2  13071  hashge2el2dif  13117  lssne0  18772  qtopeu  21329  rngoueqz  32909  mapdm0  38378  lmod0rng  41658