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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 2153 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 2153 | . . 3 ⊢ (𝑢 ∈ 𝑣 ↔ ∃𝑤(𝑤 = 𝑢 ∧ 𝑤 ∈ 𝑣)) | |
2 | 1 | gen2 1871 | . 2 ⊢ ∀𝑢∀𝑣(𝑢 ∈ 𝑣 ↔ ∃𝑤(𝑤 = 𝑢 ∧ 𝑤 ∈ 𝑣)) |
3 | 2 | bj-df-clel 33217 | 1 ⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∧ wa 382 = wceq 1631 ∃wex 1852 ∈ wcel 2145 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 |
This theorem depends on definitions: df-bi 197 df-an 383 df-ex 1853 df-clel 2767 |
This theorem is referenced by: (None) |
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