Theorem mosn 3406

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 3486 and snprc 3435. (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 2716	. 2 ⊢ ∃*𝑥 𝑥 = 𝐴
2		velsn 3392	. . 3 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
3	2	mobii 1937	. 2 ⊢ (∃*𝑥 𝑥 ∈ {𝐴} ↔ ∃*𝑥 𝑥 = 𝐴)
4	1, 3	mpbir 134	1 ⊢ ∃*𝑥 𝑥 ∈ {𝐴}

Syntax hints:   = wceq 1243   ∈ wcel 1393  ∃*wmo 1901  {csn 3375

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-sn 3381

This theorem is referenced by: (None)