Theorem bj-snexg 37778

Description: A singleton built on a set is a set. Contrary to bj-snex 37779, this proof is intuitionistically valid and does not require ax-nul 5263. (Contributed by NM, 7-Aug-1994.) Extract it from snex 5404 and prove it from ax-bj-sn 37777. (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 4594	. . 3 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
2		ax-bj-sn 37777	. . . . 5 ⊢ ∀𝑥∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ 𝑧 = 𝑥)
3	2	spi 2220	. . . 4 ⊢ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ 𝑧 = 𝑥)
4		bj-axsn 37776	. . . 4 ⊢ ({𝑥} ∈ V ↔ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ 𝑧 = 𝑥))
5	3, 4	mpbir 234	. . 3 ⊢ {𝑥} ∈ V
6	1, 5	eqeltrrdi 2869	. 2 ⊢ (𝑥 = 𝐴 → {𝐴} ∈ V)
7	6	vtocleg 3516	1 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ V)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wal 1568   = wceq 1570  ∃wex 1812   ∈ wcel 2145  Vcvv 3450  {csn 4584

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-12 2213  ax-ext 2732  ax-bj-sn 37777

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-sn 4585

This theorem is used by:  bj-snex  37779  bj-prexg  37783  bj-adjfrombun  37790