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Theorem eqsnm 3607
Description: Two ways to express that an inhabited set equals a singleton. (Contributed by Jim Kingdon, 11-Aug-2018.)
Assertion
Ref Expression
eqsnm (∃𝑥 𝑥𝐴 → (𝐴 = {𝐵} ↔ ∀𝑥𝐴 𝑥 = 𝐵))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem eqsnm
StepHypRef Expression
1 dfss3 3018 . . 3 (𝐴 ⊆ {𝐵} ↔ ∀𝑥𝐴 𝑥 ∈ {𝐵})
2 velsn 3469 . . . 4 (𝑥 ∈ {𝐵} ↔ 𝑥 = 𝐵)
32ralbii 2385 . . 3 (∀𝑥𝐴 𝑥 ∈ {𝐵} ↔ ∀𝑥𝐴 𝑥 = 𝐵)
41, 3bitri 183 . 2 (𝐴 ⊆ {𝐵} ↔ ∀𝑥𝐴 𝑥 = 𝐵)
5 sssnm 3606 . 2 (∃𝑥 𝑥𝐴 → (𝐴 ⊆ {𝐵} ↔ 𝐴 = {𝐵}))
64, 5syl5rbbr 194 1 (∃𝑥 𝑥𝐴 → (𝐴 = {𝐵} ↔ ∀𝑥𝐴 𝑥 = 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104   = wceq 1290  wex 1427  wcel 1439  wral 2360  wss 3002  {csn 3452
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 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071
This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-v 2624  df-in 3008  df-ss 3015  df-sn 3458
This theorem is referenced by:  nninfall  12203
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