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Theorem sssnm 3620
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 3033 . . . . . . . . . 10 (𝐴 ⊆ {𝐵} → (𝑥𝐴𝑥 ∈ {𝐵}))
2 elsni 3484 . . . . . . . . . 10 (𝑥 ∈ {𝐵} → 𝑥 = 𝐵)
31, 2syl6 33 . . . . . . . . 9 (𝐴 ⊆ {𝐵} → (𝑥𝐴𝑥 = 𝐵))
4 eleq1 2157 . . . . . . . . 9 (𝑥 = 𝐵 → (𝑥𝐴𝐵𝐴))
53, 4syl6 33 . . . . . . . 8 (𝐴 ⊆ {𝐵} → (𝑥𝐴 → (𝑥𝐴𝐵𝐴)))
65ibd 177 . . . . . . 7 (𝐴 ⊆ {𝐵} → (𝑥𝐴𝐵𝐴))
76exlimdv 1754 . . . . . 6 (𝐴 ⊆ {𝐵} → (∃𝑥 𝑥𝐴𝐵𝐴))
8 snssi 3603 . . . . . 6 (𝐵𝐴 → {𝐵} ⊆ 𝐴)
97, 8syl6 33 . . . . 5 (𝐴 ⊆ {𝐵} → (∃𝑥 𝑥𝐴 → {𝐵} ⊆ 𝐴))
109anc2li 323 . . . 4 (𝐴 ⊆ {𝐵} → (∃𝑥 𝑥𝐴 → (𝐴 ⊆ {𝐵} ∧ {𝐵} ⊆ 𝐴)))
11 eqss 3054 . . . 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 3093 . 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 1296  wex 1433  wcel 1445  wss 3013  {csn 3466
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077
This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-v 2635  df-in 3019  df-ss 3026  df-sn 3472
This theorem is referenced by:  eqsnm  3621  exmid01  4053  exmidn0m  4054  exmidsssn  4055  exmidomni  6885  exmidsbthrlem  12633
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