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 e. _V

Assertion

Ref	Expression
unsneqsn	|- ((A u. {B}) = {C} -> (A = (/) \/ A = {B}))

Proof of Theorem unsneqsn

Step	Hyp	Ref	Expression
1		ssun2 3428	. . . . . . 7 |- {B} C_ (A u. {B})
2		unsneqsn.1	. . . . . . . 8 |- B e. _V
3	2	snid 3761	. . . . . . 7 |- B e. {B}
4	1, 3	sselii 3271	. . . . . 6 |- B e. (A u. {B})
5		eleq2 2414	. . . . . 6 |- ((A u. {B}) = {C} -> (B e. (A u. {B}) <-> B e. {C}))
6	4, 5	mpbii 202	. . . . 5 |- ((A u. {B}) = {C} -> B e. {C})
7		elsni 3758	. . . . 5 |- (B e. {C} -> B = C)
8	6, 7	syl 15	. . . 4 |- ((A u. {B}) = {C} -> B = C)
9		sneq 3745	. . . . . 6 |- (B = C -> {B} = {C})
10	9	eqeq2d 2364	. . . . 5 |- (B = C -> ((A u. {B}) = {B} <-> (A u. {B}) = {C}))
11	10	biimprd 214	. . . 4 |- (B = C -> ((A u. {B}) = {C} -> (A u. {B}) = {B}))
12	8, 11	mpcom 32	. . 3 |- ((A u. {B}) = {C} -> (A u. {B}) = {B})
13		ssequn1 3434	. . 3 |- (A C_ {B} <-> (A u. {B}) = {B})
14	12, 13	sylibr 203	. 2 |- ((A u. {B}) = {C} -> A C_ {B})
15		sssn 3865	. 2 |- (A C_ {B} <-> (A = (/) \/ A = {B}))
16	14, 15	sylib 188	1 |- ((A u. {B}) = {C} -> (A = (/) \/ A = {B}))

Syntax hints:   -> wi 4   \/ wo 357   = wceq 1642   e. wcel 1710  _Vcvv 2860   u. cun 3208   C_ 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