Theorem bj-snexg 36508

Description: A singleton built on a set is a set. Contrary to bj-snex 36509, this proof is intuitionistically valid and does not require ax-nul 5301. (Contributed by NM, 7-Aug-1994.) Extract it from snex 5428 and prove it from ax-bj-sn 36507. (Revised by BJ, 12-Jan-2025.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-snexg	⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ V)

Proof of Theorem bj-snexg

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sneq 4635	. . 3 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
2		ax-bj-sn 36507	. . . . 5 ⊢ ∀𝑥∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ 𝑧 = 𝑥)
3	2	spi 2173	. . . 4 ⊢ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ 𝑧 = 𝑥)
4		bj-axsn 36506	. . . 4 ⊢ ({𝑥} ∈ V ↔ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ 𝑧 = 𝑥))
5	3, 4	mpbir 230	. . 3 ⊢ {𝑥} ∈ V
6	1, 5	eqeltrrdi 2838	. 2 ⊢ (𝑥 = 𝐴 → {𝐴} ∈ V)
7	6	vtocleg 3538	1 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ V)

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1532   = wceq 1534  ∃wex 1774   ∈ wcel 2099  Vcvv 3470  {csn 4625

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-12 2167  ax-ext 2699  ax-bj-sn 36507

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-v 3472  df-sn 4626

This theorem is referenced by:  bj-snex  36509  bj-prexg  36513  bj-adjfrombun  36520