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Theorem sssnm 3769
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 3164 . . . . . . . . . 10 (𝐴 ⊆ {𝐵} → (𝑥𝐴𝑥 ∈ {𝐵}))
2 elsni 3625 . . . . . . . . . 10 (𝑥 ∈ {𝐵} → 𝑥 = 𝐵)
31, 2syl6 33 . . . . . . . . 9 (𝐴 ⊆ {𝐵} → (𝑥𝐴𝑥 = 𝐵))
4 eleq1 2252 . . . . . . . . 9 (𝑥 = 𝐵 → (𝑥𝐴𝐵𝐴))
53, 4syl6 33 . . . . . . . 8 (𝐴 ⊆ {𝐵} → (𝑥𝐴 → (𝑥𝐴𝐵𝐴)))
65ibd 178 . . . . . . 7 (𝐴 ⊆ {𝐵} → (𝑥𝐴𝐵𝐴))
76exlimdv 1830 . . . . . 6 (𝐴 ⊆ {𝐵} → (∃𝑥 𝑥𝐴𝐵𝐴))
8 snssi 3751 . . . . . 6 (𝐵𝐴 → {𝐵} ⊆ 𝐴)
97, 8syl6 33 . . . . 5 (𝐴 ⊆ {𝐵} → (∃𝑥 𝑥𝐴 → {𝐵} ⊆ 𝐴))
109anc2li 329 . . . 4 (𝐴 ⊆ {𝐵} → (∃𝑥 𝑥𝐴 → (𝐴 ⊆ {𝐵} ∧ {𝐵} ⊆ 𝐴)))
11 eqss 3185 . . . 4 (𝐴 = {𝐵} ↔ (𝐴 ⊆ {𝐵} ∧ {𝐵} ⊆ 𝐴))
1210, 11imbitrrdi 162 . . 3 (𝐴 ⊆ {𝐵} → (∃𝑥 𝑥𝐴𝐴 = {𝐵}))
1312com12 30 . 2 (∃𝑥 𝑥𝐴 → (𝐴 ⊆ {𝐵} → 𝐴 = {𝐵}))
14 eqimss 3224 . 2 (𝐴 = {𝐵} → 𝐴 ⊆ {𝐵})
1513, 14impbid1 142 1 (∃𝑥 𝑥𝐴 → (𝐴 ⊆ {𝐵} ↔ 𝐴 = {𝐵}))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1364  wex 1503  wcel 2160  wss 3144  {csn 3607
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-in 3150  df-ss 3157  df-sn 3613
This theorem is referenced by:  eqsnm  3770  ss1o0el1  4215  exmidn0m  4219  exmidsssn  4220  exmidomni  7171  exmidunben  12480  exmidsbthrlem  15249  sbthomlem  15252
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