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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 1985 could be relabeled as 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 1985 | . 2 ⊢ (𝑢 ∈ 𝑣 ↔ ∃𝑤(𝑤 = 𝑢 ∧ 𝑤 ∈ 𝑣)) | |
2 | 1 | bj-df-clel 32081 | 1 ⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 = wceq 1475 ∃wex 1695 ∈ wcel 1977 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 |
This theorem depends on definitions: df-bi 196 df-an 385 df-ex 1696 df-clel 2606 |
This theorem is referenced by: (None) |
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