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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 3561 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴}) | |
2 | bj-snexg 13281 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ V) | |
3 | sbcel2g 3028 | . . . . 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 3035 | . . . . . 6 ⊢ ({𝐴} ∈ V → ⦋{𝐴} / 𝑥⦌𝑥 = {𝐴}) | |
6 | 2, 5 | syl 14 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → ⦋{𝐴} / 𝑥⦌𝑥 = {𝐴}) |
7 | 6 | eleq2d 2210 | . . . 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 2999 | 1 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝐴 ∈ 𝑥) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1332 ∃wex 1469 ∈ wcel 1481 Vcvv 2689 [wsbc 2913 ⦋csb 3007 {csn 3532 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-pr 4139 ax-bdor 13185 ax-bdeq 13189 ax-bdsep 13253 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-rex 2423 df-v 2691 df-sbc 2914 df-csb 3008 df-un 3080 df-sn 3538 df-pr 3539 |
This theorem is referenced by: (None) |
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