Theorem eqsn 3867

Description: Two ways to express that a nonempty set equals a singleton. (Contributed by NM, 15-Dec-2007.)

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		eqimss 3323	. . 3 ⊢ (A = {B} → A ⊆ {B})
2		df-ne 2518	. . . . 5 ⊢ (A ≠ ∅ ↔ ¬ A = ∅)
3		sssn 3864	. . . . . . 7 ⊢ (A ⊆ {B} ↔ (A = ∅ ∨ A = {B}))
4	3	biimpi 186	. . . . . 6 ⊢ (A ⊆ {B} → (A = ∅ ∨ A = {B}))
5	4	ord 366	. . . . 5 ⊢ (A ⊆ {B} → (¬ A = ∅ → A = {B}))
6	2, 5	syl5bi 208	. . . 4 ⊢ (A ⊆ {B} → (A ≠ ∅ → A = {B}))
7	6	com12 27	. . 3 ⊢ (A ≠ ∅ → (A ⊆ {B} → A = {B}))
8	1, 7	impbid2 195	. 2 ⊢ (A ≠ ∅ → (A = {B} ↔ A ⊆ {B}))
9		dfss3 3263	. . 3 ⊢ (A ⊆ {B} ↔ ∀x ∈ A x ∈ {B})
10		elsn 3748	. . . 4 ⊢ (x ∈ {B} ↔ x = B)
11	10	ralbii 2638	. . 3 ⊢ (∀x ∈ A x ∈ {B} ↔ ∀x ∈ A x = B)
12	9, 11	bitri 240	. 2 ⊢ (A ⊆ {B} ↔ ∀x ∈ A x = B)
13	8, 12	syl6bb 252	1 ⊢ (A ≠ ∅ → (A = {B} ↔ ∀x ∈ A x = B))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∨ wo 357   = wceq 1642   ∈ wcel 1710   ≠ wne 2516  ∀wral 2614   ⊆ wss 3257  ∅c0 3550  {csn 3737

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-ss 3259  df-nul 3551  df-sn 3741

This theorem is referenced by: (None)