Theorem mosn 3674

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 3756 and snprc 3703. (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 2952	. 2 ⊢ ∃*𝑥 𝑥 = 𝐴
2		velsn 3655	. . 3 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
3	2	mobii 2092	. 2 ⊢ (∃*𝑥 𝑥 ∈ {𝐴} ↔ ∃*𝑥 𝑥 = 𝐴)
4	1, 3	mpbir 146	1 ⊢ ∃*𝑥 𝑥 ∈ {𝐴}

Syntax hints:   = wceq 1373  ∃*wmo 2056   ∈ wcel 2177  {csn 3638

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-sn 3644

This theorem is referenced by: (None)