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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 3524 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴}) | |
2 | bj-snexg 13037 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ V) | |
3 | sbcel2g 2994 | . . . . 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 3000 | . . . . . 6 ⊢ ({𝐴} ∈ V → ⦋{𝐴} / 𝑥⦌𝑥 = {𝐴}) | |
6 | 2, 5 | syl 14 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → ⦋{𝐴} / 𝑥⦌𝑥 = {𝐴}) |
7 | 6 | eleq2d 2187 | . . . 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 2967 | 1 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝐴 ∈ 𝑥) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1316 ∃wex 1453 ∈ wcel 1465 Vcvv 2660 [wsbc 2882 ⦋csb 2975 {csn 3497 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-pr 4101 ax-bdor 12941 ax-bdeq 12945 ax-bdsep 13009 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-rex 2399 df-v 2662 df-sbc 2883 df-csb 2976 df-un 3045 df-sn 3503 df-pr 3504 |
This theorem is referenced by: (None) |
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