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Theorem sssnm 3567
 Description: The inhabited subset of a singleton. (Contributed by Jim Kingdon, 10-Aug-2018.)
Assertion
Ref Expression
sssnm (∃𝑥 𝑥𝐴 → (𝐴 ⊆ {𝐵} ↔ 𝐴 = {𝐵}))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem sssnm
StepHypRef Expression
1 ssel 3003 . . . . . . . . . 10 (𝐴 ⊆ {𝐵} → (𝑥𝐴𝑥 ∈ {𝐵}))
2 elsni 3435 . . . . . . . . . 10 (𝑥 ∈ {𝐵} → 𝑥 = 𝐵)
31, 2syl6 33 . . . . . . . . 9 (𝐴 ⊆ {𝐵} → (𝑥𝐴𝑥 = 𝐵))
4 eleq1 2145 . . . . . . . . 9 (𝑥 = 𝐵 → (𝑥𝐴𝐵𝐴))
53, 4syl6 33 . . . . . . . 8 (𝐴 ⊆ {𝐵} → (𝑥𝐴 → (𝑥𝐴𝐵𝐴)))
65ibd 176 . . . . . . 7 (𝐴 ⊆ {𝐵} → (𝑥𝐴𝐵𝐴))
76exlimdv 1742 . . . . . 6 (𝐴 ⊆ {𝐵} → (∃𝑥 𝑥𝐴𝐵𝐴))
8 snssi 3550 . . . . . 6 (𝐵𝐴 → {𝐵} ⊆ 𝐴)
97, 8syl6 33 . . . . 5 (𝐴 ⊆ {𝐵} → (∃𝑥 𝑥𝐴 → {𝐵} ⊆ 𝐴))
109anc2li 322 . . . 4 (𝐴 ⊆ {𝐵} → (∃𝑥 𝑥𝐴 → (𝐴 ⊆ {𝐵} ∧ {𝐵} ⊆ 𝐴)))
11 eqss 3024 . . . 4 (𝐴 = {𝐵} ↔ (𝐴 ⊆ {𝐵} ∧ {𝐵} ⊆ 𝐴))
1210, 11syl6ibr 160 . . 3 (𝐴 ⊆ {𝐵} → (∃𝑥 𝑥𝐴𝐴 = {𝐵}))
1312com12 30 . 2 (∃𝑥 𝑥𝐴 → (𝐴 ⊆ {𝐵} → 𝐴 = {𝐵}))
14 eqimss 3061 . 2 (𝐴 = {𝐵} → 𝐴 ⊆ {𝐵})
1513, 14impbid1 140 1 (∃𝑥 𝑥𝐴 → (𝐴 ⊆ {𝐵} ↔ 𝐴 = {𝐵}))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103   = wceq 1285  ∃wex 1422   ∈ wcel 1434   ⊆ wss 2983  {csn 3417 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065 This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612  df-in 2989  df-ss 2996  df-sn 3423 This theorem is referenced by:  eqsnm  3568  exmid01  3989
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