Theorem bj-axsn 37027

Description: Two ways of stating the axiom of singleton (which is the universal closure of either side, see ax-bj-sn 37028). (Contributed by BJ, 12-Jan-2025.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-axsn	⊢ ({𝑥} ∈ V ↔ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ 𝑧 = 𝑥))

Distinct variable group:   𝑥,𝑦,𝑧

Proof of Theorem bj-axsn

Step	Hyp	Ref	Expression
1		velsn 4608	. 2 ⊢ (𝑧 ∈ {𝑥} ↔ 𝑧 = 𝑥)
2	1	bj-clex 37026	1 ⊢ ({𝑥} ∈ V ↔ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ 𝑧 = 𝑥))

Syntax hints:   ↔ wb 206  ∀wal 1538  ∃wex 1779   ∈ wcel 2109  Vcvv 3450  {csn 4592

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-v 3452  df-sn 4593

This theorem is referenced by:  bj-snexg  37029