Theorem bj-axsn 37033

Description: Two ways of stating the axiom of singleton (which is the universal closure of either side, see ax-bj-sn 37034). (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 4642	. 2 ⊢ (𝑧 ∈ {𝑥} ↔ 𝑧 = 𝑥)
2	1	bj-clex 37032	1 ⊢ ({𝑥} ∈ V ↔ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ 𝑧 = 𝑥))

Syntax hints:   ↔ wb 206  ∀wal 1538  ∃wex 1779   ∈ wcel 2108  Vcvv 3480  {csn 4626

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 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-sn 4627

This theorem is referenced by:  bj-snexg  37035