Theorem bj-sels 32950

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 4206	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴})
2		sbcel2 3989	. . . 4 ⊢ ([{𝐴} / 𝑥]𝐴 ∈ 𝑥 ↔ 𝐴 ∈ ⦋{𝐴} / 𝑥⦌𝑥)
3		snex 4908	. . . . . 6 ⊢ {𝐴} ∈ V
4		csbvarg 4003	. . . . . 6 ⊢ ({𝐴} ∈ V → ⦋{𝐴} / 𝑥⦌𝑥 = {𝐴})
5	3, 4	ax-mp 5	. . . . 5 ⊢ ⦋{𝐴} / 𝑥⦌𝑥 = {𝐴}
6	5	eleq2i 2693	. . . 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 3522	1 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝐴 ∈ 𝑥)

Syntax hints:   → wi 4   = wceq 1483  ∃wex 1704   ∈ wcel 1990  Vcvv 3200  [wsbc 3435  ⦋csb 3533  {csn 4177

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-nul 3916  df-sn 4178  df-pr 4180

This theorem is referenced by: (None)