Theorem bj-snexg 37554

Description: A singleton built on a set is a set. Contrary to bj-snex 37555, this proof is intuitionistically valid and does not require ax-nul 5268. (Contributed by NM, 7-Aug-1994.) Extract it from snex 5408 and prove it from ax-bj-sn 37553. (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 4601	. . 3 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
2		ax-bj-sn 37553	. . . . 5 ⊢ ∀𝑥∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ 𝑧 = 𝑥)
3	2	spi 2226	. . . 4 ⊢ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ 𝑧 = 𝑥)
4		bj-axsn 37552	. . . 4 ⊢ ({𝑥} ∈ V ↔ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ 𝑧 = 𝑥))
5	3, 4	mpbir 234	. . 3 ⊢ {𝑥} ∈ V
6	1, 5	eqeltrrdi 2878	. 2 ⊢ (𝑥 = 𝐴 → {𝐴} ∈ V)
7	6	vtocleg 3530	1 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ V)

Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1565   = wceq 1567  ∃wex 1806   ∈ wcel 2149  Vcvv 3463  {csn 4591

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-12 2219  ax-ext 2741  ax-bj-sn 37553

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-v 3465  df-sn 4592

This theorem is referenced by:  bj-snex  37555  bj-prexg  37559  bj-adjfrombun  37566