Theorem bj-axsn 37696

Description: Two ways of stating the axiom of singleton (which is the universal closure of either side, see ax-bj-sn 37697). (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 4604	. 2 ⊢ (𝑧 ∈ {𝑥} ↔ 𝑧 = 𝑥)
2	1	bj-clex 37695	1 ⊢ ({𝑥} ∈ V ↔ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ 𝑧 = 𝑥))

This proof depends on syntax axioms:   ↔ wb 209  ∀wal 1567  ∃wex 1808   ∈ wcel 2142  Vcvv 3454  {csn 4588

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-tru 1572  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3456  df-sn 4589

This theorem is used by:  bj-snexg  37698