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| Mirrors > Home > MPE Home > Th. List > Mathboxes > rp-abid | Structured version Visualization version GIF version | ||
| Description: Two ways to express a class. (Contributed by RP, 13-Feb-2025.) |
| Ref | Expression |
|---|---|
| rp-abid | ⊢ 𝐴 = {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = 𝑎} |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | clel5 3665 | . 2 ⊢ (𝑥 ∈ 𝐴 ↔ ∃𝑎 ∈ 𝐴 𝑥 = 𝑎) | |
| 2 | 1 | eqabi 2877 | 1 ⊢ 𝐴 = {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = 𝑎} |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 {cab 2714 ∃wrex 3070 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rex 3071 |
| This theorem is referenced by: oaun2 43394 oaun3 43395 |
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