Theorem bj-snexg 37480

Description: A singleton built on a set is a set. Contrary to bj-snex 37481, this proof is intuitionistically valid and does not require ax-nul 5253. (Contributed by NM, 7-Aug-1994.) Extract it from snex 5393 and prove it from ax-bj-sn 37479. (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 4589	. . 3 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
2		ax-bj-sn 37479	. . . . 5 ⊢ ∀𝑥∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ 𝑧 = 𝑥)
3	2	spi 2218	. . . 4 ⊢ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ 𝑧 = 𝑥)
4		bj-axsn 37478	. . . 4 ⊢ ({𝑥} ∈ V ↔ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ 𝑧 = 𝑥))
5	3, 4	mpbir 233	. . 3 ⊢ {𝑥} ∈ V
6	1, 5	eqeltrrdi 2870	. 2 ⊢ (𝑥 = 𝐴 → {𝐴} ∈ V)
7	6	vtocleg 3520	1 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ V)

Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1557   = wceq 1559  ∃wex 1798   ∈ wcel 2141  Vcvv 3453  {csn 4579

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-12 2211  ax-ext 2733  ax-bj-sn 37479

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1562  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3455  df-sn 4580

This theorem is referenced by:  bj-snex  37481  bj-prexg  37485  bj-adjfrombun  37492