Theorem bj-axsn 36642

Description: Two ways of stating the axiom of singleton (which is the universal closure of either side, see ax-bj-sn 36643). (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 4646	. 2 ⊢ (𝑧 ∈ {𝑥} ↔ 𝑧 = 𝑥)
2	1	bj-clex 36641	1 ⊢ ({𝑥} ∈ V ↔ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ 𝑧 = 𝑥))

Syntax hints:   ↔ wb 205  ∀wal 1531  ∃wex 1773   ∈ wcel 2098  Vcvv 3461  {csn 4630

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-v 3463  df-sn 4631

This theorem is referenced by:  bj-snexg  36644