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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
Assertion
Ref Expression
unsneqsn

Proof of Theorem unsneqsn
StepHypRef Expression
1 ssun2 3427 . . . . . . 7
2 unsneqsn.1 . . . . . . . 8
32snid 3760 . . . . . . 7
41, 3sselii 3270 . . . . . 6
5 eleq2 2414 . . . . . 6
64, 5mpbii 202 . . . . 5
7 elsni 3757 . . . . 5
86, 7syl 15 . . . 4
9 sneq 3744 . . . . . 6
109eqeq2d 2364 . . . . 5
1110biimprd 214 . . . 4
128, 11mpcom 32 . . 3
13 ssequn1 3433 . . 3
1412, 13sylibr 203 . 2
15 sssn 3864 . 2
1614, 15sylib 188 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wo 357   wceq 1642   wcel 1710  cvv 2859   cun 3207   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-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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