Theorem mosn 3658

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 3740 and snprc 3687. (Contributed by Jim Kingdon, 30-Aug-2018.)

Assertion

Ref	Expression
mosn	⊢ ∃*𝑥 𝑥 ∈ {𝐴}

Distinct variable group:   𝑥,𝐴

Proof of Theorem mosn

Step	Hyp	Ref	Expression
1		moeq 2939	. 2 ⊢ ∃*𝑥 𝑥 = 𝐴
2		velsn 3639	. . 3 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
3	2	mobii 2082	. 2 ⊢ (∃*𝑥 𝑥 ∈ {𝐴} ↔ ∃*𝑥 𝑥 = 𝐴)
4	1, 3	mpbir 146	1 ⊢ ∃*𝑥 𝑥 ∈ {𝐴}

Syntax hints:   = wceq 1364  ∃*wmo 2046   ∈ wcel 2167  {csn 3622

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-sn 3628

This theorem is referenced by: (None)