Theorem bj-snexg 36219

Description: A singleton built on a set is a set. Contrary to bj-snex 36220, this proof is intuitionistically valid and does not require ax-nul 5306. (Contributed by NM, 7-Aug-1994.) Extract it from snex 5431 and prove it from ax-bj-sn 36218. (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 4638	. . 3 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
2		ax-bj-sn 36218	. . . . 5 ⊢ ∀𝑥∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ 𝑧 = 𝑥)
3	2	spi 2176	. . . 4 ⊢ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ 𝑧 = 𝑥)
4		bj-axsn 36217	. . . 4 ⊢ ({𝑥} ∈ V ↔ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ 𝑧 = 𝑥))
5	3, 4	mpbir 230	. . 3 ⊢ {𝑥} ∈ V
6	1, 5	eqeltrrdi 2841	. 2 ⊢ (𝑥 = 𝐴 → {𝐴} ∈ V)
7	6	vtocleg 3541	1 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ V)

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1538   = wceq 1540  ∃wex 1780   ∈ wcel 2105  Vcvv 3473  {csn 4628

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-12 2170  ax-ext 2702  ax-bj-sn 36218

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-v 3475  df-sn 4629

This theorem is referenced by:  bj-snex  36220  bj-prexg  36224  bj-adjfrombun  36231