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Theorem sssnm 3499
 Description: The inhabited subset of a singleton. (Contributed by Jim Kingdon, 10-Aug-2018.)
Assertion
Ref Expression
sssnm (x x A → (A ⊆ {B} ↔ A = {B}))
Distinct variable groups:   x,A   x,B

Proof of Theorem sssnm
StepHypRef Expression
1 ssel 2916 . . . . . . . . . 10 (A ⊆ {B} → (x Ax {B}))
2 elsni 3374 . . . . . . . . . 10 (x {B} → x = B)
31, 2syl6 29 . . . . . . . . 9 (A ⊆ {B} → (x Ax = B))
4 eleq1 2082 . . . . . . . . 9 (x = B → (x AB A))
53, 4syl6 29 . . . . . . . 8 (A ⊆ {B} → (x A → (x AB A)))
65ibd 167 . . . . . . 7 (A ⊆ {B} → (x AB A))
76exlimdv 1682 . . . . . 6 (A ⊆ {B} → (x x AB A))
8 snssi 3482 . . . . . 6 (B A → {B} ⊆ A)
97, 8syl6 29 . . . . 5 (A ⊆ {B} → (x x A → {B} ⊆ A))
109anc2li 312 . . . 4 (A ⊆ {B} → (x x A → (A ⊆ {B} {B} ⊆ A)))
11 eqss 2937 . . . 4 (A = {B} ↔ (A ⊆ {B} {B} ⊆ A))
1210, 11syl6ibr 151 . . 3 (A ⊆ {B} → (x x AA = {B}))
1312com12 27 . 2 (x x A → (A ⊆ {B} → A = {B}))
14 eqimss 2974 . 2 (A = {B} → A ⊆ {B})
1513, 14impbid1 130 1 (x x A → (A ⊆ {B} ↔ A = {B}))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1228  ∃wex 1362   ∈ wcel 1374   ⊆ wss 2894  {csn 3350 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004 This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-in 2901  df-ss 2908  df-sn 3356 This theorem is referenced by:  eqsnm  3500
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