Theorem bj-axsn 37076

Description: Two ways of stating the axiom of singleton (which is the universal closure of either side, see ax-bj-sn 37077). (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 4589	. 2 ⊢ (𝑧 ∈ {𝑥} ↔ 𝑧 = 𝑥)
2	1	bj-clex 37075	1 ⊢ ({𝑥} ∈ V ↔ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ 𝑧 = 𝑥))

Syntax hints:   ↔ wb 206  ∀wal 1539  ∃wex 1780   ∈ wcel 2111  Vcvv 3436  {csn 4573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-v 3438  df-sn 4574

This theorem is referenced by:  bj-snexg  37078