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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 3423 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴}) | |
2 | bj-snexg 10703 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ V) | |
3 | sbcel2g 2927 | . . . . 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 2933 | . . . . . 6 ⊢ ({𝐴} ∈ V → ⦋{𝐴} / 𝑥⦌𝑥 = {𝐴}) | |
6 | 2, 5 | syl 14 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → ⦋{𝐴} / 𝑥⦌𝑥 = {𝐴}) |
7 | 6 | eleq2d 2148 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ⦋{𝐴} / 𝑥⦌𝑥 ↔ 𝐴 ∈ {𝐴})) |
8 | 4, 7 | bitrd 186 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ([{𝐴} / 𝑥]𝐴 ∈ 𝑥 ↔ 𝐴 ∈ {𝐴})) |
9 | 1, 8 | mpbird 165 | . 2 ⊢ (𝐴 ∈ 𝑉 → [{𝐴} / 𝑥]𝐴 ∈ 𝑥) |
10 | 9 | spesbcd 2900 | 1 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝐴 ∈ 𝑥) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 103 = wceq 1284 ∃wex 1421 ∈ wcel 1433 Vcvv 2601 [wsbc 2815 ⦋csb 2908 {csn 3398 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-pr 3964 ax-bdor 10607 ax-bdeq 10611 ax-bdsep 10675 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-rex 2354 df-v 2603 df-sbc 2816 df-csb 2909 df-un 2977 df-sn 3404 df-pr 3405 |
This theorem is referenced by: (None) |
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