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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-sels | GIF version |
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.) |
Ref | Expression |
---|---|
bj-sels | ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝐴 ∈ 𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snidg 3589 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴}) | |
2 | bj-snexg 13498 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ V) | |
3 | sbcel2g 3052 | . . . . 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 3059 | . . . . . 6 ⊢ ({𝐴} ∈ V → ⦋{𝐴} / 𝑥⦌𝑥 = {𝐴}) | |
6 | 2, 5 | syl 14 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → ⦋{𝐴} / 𝑥⦌𝑥 = {𝐴}) |
7 | 6 | eleq2d 2227 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ⦋{𝐴} / 𝑥⦌𝑥 ↔ 𝐴 ∈ {𝐴})) |
8 | 4, 7 | bitrd 187 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ([{𝐴} / 𝑥]𝐴 ∈ 𝑥 ↔ 𝐴 ∈ {𝐴})) |
9 | 1, 8 | mpbird 166 | . 2 ⊢ (𝐴 ∈ 𝑉 → [{𝐴} / 𝑥]𝐴 ∈ 𝑥) |
10 | 9 | spesbcd 3023 | 1 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝐴 ∈ 𝑥) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1335 ∃wex 1472 ∈ wcel 2128 Vcvv 2712 [wsbc 2937 ⦋csb 3031 {csn 3560 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-pr 4169 ax-bdor 13402 ax-bdeq 13406 ax-bdsep 13470 |
This theorem depends on definitions: df-bi 116 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-rex 2441 df-v 2714 df-sbc 2938 df-csb 3032 df-un 3106 df-sn 3566 df-pr 3567 |
This theorem is referenced by: (None) |
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