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Theorem sssnm 3651
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 3061 . . . . . . . . . 10 (𝐴 ⊆ {𝐵} → (𝑥𝐴𝑥 ∈ {𝐵}))
2 elsni 3515 . . . . . . . . . 10 (𝑥 ∈ {𝐵} → 𝑥 = 𝐵)
31, 2syl6 33 . . . . . . . . 9 (𝐴 ⊆ {𝐵} → (𝑥𝐴𝑥 = 𝐵))
4 eleq1 2180 . . . . . . . . 9 (𝑥 = 𝐵 → (𝑥𝐴𝐵𝐴))
53, 4syl6 33 . . . . . . . 8 (𝐴 ⊆ {𝐵} → (𝑥𝐴 → (𝑥𝐴𝐵𝐴)))
65ibd 177 . . . . . . 7 (𝐴 ⊆ {𝐵} → (𝑥𝐴𝐵𝐴))
76exlimdv 1775 . . . . . 6 (𝐴 ⊆ {𝐵} → (∃𝑥 𝑥𝐴𝐵𝐴))
8 snssi 3634 . . . . . 6 (𝐵𝐴 → {𝐵} ⊆ 𝐴)
97, 8syl6 33 . . . . 5 (𝐴 ⊆ {𝐵} → (∃𝑥 𝑥𝐴 → {𝐵} ⊆ 𝐴))
109anc2li 327 . . . 4 (𝐴 ⊆ {𝐵} → (∃𝑥 𝑥𝐴 → (𝐴 ⊆ {𝐵} ∧ {𝐵} ⊆ 𝐴)))
11 eqss 3082 . . . 4 (𝐴 = {𝐵} ↔ (𝐴 ⊆ {𝐵} ∧ {𝐵} ⊆ 𝐴))
1210, 11syl6ibr 161 . . 3 (𝐴 ⊆ {𝐵} → (∃𝑥 𝑥𝐴𝐴 = {𝐵}))
1312com12 30 . 2 (∃𝑥 𝑥𝐴 → (𝐴 ⊆ {𝐵} → 𝐴 = {𝐵}))
14 eqimss 3121 . 2 (𝐴 = {𝐵} → 𝐴 ⊆ {𝐵})
1513, 14impbid1 141 1 (∃𝑥 𝑥𝐴 → (𝐴 ⊆ {𝐵} ↔ 𝐴 = {𝐵}))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1316  wex 1453  wcel 1465  wss 3041  {csn 3497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-in 3047  df-ss 3054  df-sn 3503
This theorem is referenced by:  eqsnm  3652  exmid01  4091  exmidn0m  4094  exmidsssn  4095  exmidomni  6982  exmidunben  11866  exmidsbthrlem  13144  sbthomlem  13147
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