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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-dfclel | Structured version Visualization version GIF version |
Description: Characterization of the elements of a class. Note: cleljust 1998 could be relabeled "clelhyp". (Contributed by BJ, 27-Jun-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-dfclel | ⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cleljust 1998 | . . 3 ⊢ (𝑢 ∈ 𝑣 ↔ ∃𝑤(𝑤 = 𝑢 ∧ 𝑤 ∈ 𝑣)) | |
2 | 1 | gen2 1723 | . 2 ⊢ ∀𝑢∀𝑣(𝑢 ∈ 𝑣 ↔ ∃𝑤(𝑤 = 𝑢 ∧ 𝑤 ∈ 𝑣)) |
3 | 2 | bj-df-clel 32888 | 1 ⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∧ wa 384 = wceq 1483 ∃wex 1704 ∈ wcel 1990 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 |
This theorem depends on definitions: df-bi 197 df-an 386 df-ex 1705 df-clel 2618 |
This theorem is referenced by: (None) |
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