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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
StepHypRef 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}))
43biimpi 186 . . . . . 6 (A {B} → (A = A = {B}))
54ord 366 . . . . 5 (A {B} → (¬ A = A = {B}))
62, 5syl5bi 208 . . . 4 (A {B} → (AA = {B}))
76com12 27 . . 3 (A → (A {B} → A = {B}))
81, 7impbid2 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)
1110ralbii 2638 . . 3 (x A x {B} ↔ x A x = B)
129, 11bitri 240 . 2 (A {B} ↔ x A x = B)
138, 12syl6bb 252 1 (A → (A = {B} ↔ x A x = B))
 Colors of variables: wff setvar class 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-1 5  ax-2 6  ax-3 7  ax-mp 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)
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