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Theorem mosn 3434
Description: A singleton has at most one element. This works whether 𝐴 is a proper class or not, and in that sense can be seen as encompassing both snmg 3514 and snprc 3463. (Contributed by Jim Kingdon, 30-Aug-2018.)
Assertion
Ref Expression
mosn ∃*𝑥 𝑥 ∈ {𝐴}
Distinct variable group:   𝑥,𝐴

Proof of Theorem mosn
StepHypRef Expression
1 moeq 2739 . 2 ∃*𝑥 𝑥 = 𝐴
2 velsn 3420 . . 3 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
32mobii 1953 . 2 (∃*𝑥 𝑥 ∈ {𝐴} ↔ ∃*𝑥 𝑥 = 𝐴)
41, 3mpbir 138 1 ∃*𝑥 𝑥 ∈ {𝐴}
Colors of variables: wff set class
Syntax hints:   = wceq 1259  wcel 1409  ∃*wmo 1917  {csn 3403
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-sn 3409
This theorem is referenced by: (None)
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