Theorem bj-sels 33256

Description: If a class is a set, then it is a member of a set. (Contributed by BJ, 3-Apr-2019.)

Assertion

Ref	Expression
bj-sels	⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝐴 ∈ 𝑥)

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem bj-sels

Step	Hyp	Ref	Expression
1		snidg 4351	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴})
2		sbcel2 4132	. . . 4 ⊢ ([{𝐴} / 𝑥]𝐴 ∈ 𝑥 ↔ 𝐴 ∈ ⦋{𝐴} / 𝑥⦌𝑥)
3		snex 5057	. . . . . 6 ⊢ {𝐴} ∈ V
4		csbvarg 4146	. . . . . 6 ⊢ ({𝐴} ∈ V → ⦋{𝐴} / 𝑥⦌𝑥 = {𝐴})
5	3, 4	ax-mp 5	. . . . 5 ⊢ ⦋{𝐴} / 𝑥⦌𝑥 = {𝐴}
6	5	eleq2i 2831	. . . 4 ⊢ (𝐴 ∈ ⦋{𝐴} / 𝑥⦌𝑥 ↔ 𝐴 ∈ {𝐴})
7	2, 6	bitri 264	. . 3 ⊢ ([{𝐴} / 𝑥]𝐴 ∈ 𝑥 ↔ 𝐴 ∈ {𝐴})
8	1, 7	sylibr 224	. 2 ⊢ (𝐴 ∈ 𝑉 → [{𝐴} / 𝑥]𝐴 ∈ 𝑥)
9	8	spesbcd 3663	1 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝐴 ∈ 𝑥)

Syntax hints:   → wi 4   = wceq 1632  ∃wex 1853   ∈ wcel 2139  Vcvv 3340  [wsbc 3576  ⦋csb 3674  {csn 4321

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-nul 4059  df-sn 4322  df-pr 4324

This theorem is referenced by: (None)