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Theorem unsneqsn 3887
 Description: If union with a singleton yields a singleton, then the first argument is either also the singleton or is the empty set. (Contributed by SF, 15-Jan-2015.)
Hypothesis
Ref Expression
unsneqsn.1 B V
Assertion
Ref Expression
unsneqsn ((A ∪ {B}) = {C} → (A = A = {B}))

Proof of Theorem unsneqsn
StepHypRef Expression
1 ssun2 3427 . . . . . . 7 {B} (A ∪ {B})
2 unsneqsn.1 . . . . . . . 8 B V
32snid 3760 . . . . . . 7 B {B}
41, 3sselii 3270 . . . . . 6 B (A ∪ {B})
5 eleq2 2414 . . . . . 6 ((A ∪ {B}) = {C} → (B (A ∪ {B}) ↔ B {C}))
64, 5mpbii 202 . . . . 5 ((A ∪ {B}) = {C} → B {C})
7 elsni 3757 . . . . 5 (B {C} → B = C)
86, 7syl 15 . . . 4 ((A ∪ {B}) = {C} → B = C)
9 sneq 3744 . . . . . 6 (B = C → {B} = {C})
109eqeq2d 2364 . . . . 5 (B = C → ((A ∪ {B}) = {B} ↔ (A ∪ {B}) = {C}))
1110biimprd 214 . . . 4 (B = C → ((A ∪ {B}) = {C} → (A ∪ {B}) = {B}))
128, 11mpcom 32 . . 3 ((A ∪ {B}) = {C} → (A ∪ {B}) = {B})
13 ssequn1 3433 . . 3 (A {B} ↔ (A ∪ {B}) = {B})
1412, 13sylibr 203 . 2 ((A ∪ {B}) = {C} → A {B})
15 sssn 3864 . 2 (A {B} ↔ (A = A = {B}))
1614, 15sylib 188 1 ((A ∪ {B}) = {C} → (A = A = {B}))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∨ wo 357   = wceq 1642   ∈ wcel 1710  Vcvv 2859   ∪ cun 3207   ⊆ 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-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-ss 3259  df-nul 3551  df-sn 3741 This theorem is referenced by:  nnsucelr  4428
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