Theorem unsneqsn 3888

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

Step	Hyp	Ref	Expression
1		ssun2 3428	. . . . . . 7 ⊢ {B} ⊆ (A ∪ {B})
2		unsneqsn.1	. . . . . . . 8 ⊢ B ∈ V
3	2	snid 3761	. . . . . . 7 ⊢ B ∈ {B}
4	1, 3	sselii 3271	. . . . . 6 ⊢ B ∈ (A ∪ {B})
5		eleq2 2414	. . . . . 6 ⊢ ((A ∪ {B}) = {C} → (B ∈ (A ∪ {B}) ↔ B ∈ {C}))
6	4, 5	mpbii 202	. . . . 5 ⊢ ((A ∪ {B}) = {C} → B ∈ {C})
7		elsni 3758	. . . . 5 ⊢ (B ∈ {C} → B = C)
8	6, 7	syl 15	. . . 4 ⊢ ((A ∪ {B}) = {C} → B = C)
9		sneq 3745	. . . . . 6 ⊢ (B = C → {B} = {C})
10	9	eqeq2d 2364	. . . . 5 ⊢ (B = C → ((A ∪ {B}) = {B} ↔ (A ∪ {B}) = {C}))
11	10	biimprd 214	. . . 4 ⊢ (B = C → ((A ∪ {B}) = {C} → (A ∪ {B}) = {B}))
12	8, 11	mpcom 32	. . 3 ⊢ ((A ∪ {B}) = {C} → (A ∪ {B}) = {B})
13		ssequn1 3434	. . 3 ⊢ (A ⊆ {B} ↔ (A ∪ {B}) = {B})
14	12, 13	sylibr 203	. 2 ⊢ ((A ∪ {B}) = {C} → A ⊆ {B})
15		sssn 3865	. 2 ⊢ (A ⊆ {B} ↔ (A = ∅ ∨ A = {B}))
16	14, 15	sylib 188	1 ⊢ ((A ∪ {B}) = {C} → (A = ∅ ∨ A = {B}))

Syntax hints:   → wi 4   ∨ wo 357   = wceq 1642   ∈ wcel 1710  Vcvv 2860   ∪ cun 3208   ⊆ wss 3258  ∅c0 3551  {csn 3738

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 2479  df-ne 2519  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-ss 3260  df-nul 3552  df-sn 3742

This theorem is referenced by:  nnsucelr  4429