Theorem bj-sels 16857

Description: If a class is a set, then it is a member of a set. (Copied from set.mm.) (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 3737	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴})
2		bj-snexg 16855	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ V)
3		sbcel2g 3168	. . . . 5 ⊢ ({𝐴} ∈ V → ([{𝐴} / 𝑥]𝐴 ∈ 𝑥 ↔ 𝐴 ∈ ⦋{𝐴} / 𝑥⦌𝑥))
4	2, 3	syl 14	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ([{𝐴} / 𝑥]𝐴 ∈ 𝑥 ↔ 𝐴 ∈ ⦋{𝐴} / 𝑥⦌𝑥))
5		csbvarg 3175	. . . . . 6 ⊢ ({𝐴} ∈ V → ⦋{𝐴} / 𝑥⦌𝑥 = {𝐴})
6	2, 5	syl 14	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ⦋{𝐴} / 𝑥⦌𝑥 = {𝐴})
7	6	eleq2d 2308	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ⦋{𝐴} / 𝑥⦌𝑥 ↔ 𝐴 ∈ {𝐴}))
8	4, 7	bitrd 188	. . 3 ⊢ (𝐴 ∈ 𝑉 → ([{𝐴} / 𝑥]𝐴 ∈ 𝑥 ↔ 𝐴 ∈ {𝐴}))
9	1, 8	mpbird 167	. 2 ⊢ (𝐴 ∈ 𝑉 → [{𝐴} / 𝑥]𝐴 ∈ 𝑥)
10	9	spesbcd 3139	1 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝐴 ∈ 𝑥)

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1402  ∃wex 1545   ∈ wcel 2209  Vcvv 2821  [wsbc 3051  ⦋csb 3147  {csn 3708

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-pr 4344  ax-bdor 16759  ax-bdeq 16763  ax-bdsep 16827

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-sn 3714  df-pr 3715

This theorem is referenced by: (None)