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| Mirrors > Home > MPE Home > Th. List > Mathboxes > rmotru | Structured version Visualization version GIF version | ||
| Description: Two ways of expressing "at most one" element. (Contributed by Zhi Wang, 19-Sep-2024.) (Proof shortened by BJ, 23-Sep-2024.) |
| Ref | Expression |
|---|---|
| rmotru | ⊢ (∃*𝑥 𝑥 ∈ 𝐴 ↔ ∃*𝑥 ∈ 𝐴 ⊤) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | tru 1544 | . . . 4 ⊢ ⊤ | |
| 2 | 1 | biantru 529 | . . 3 ⊢ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∧ ⊤)) |
| 3 | 2 | mobii 2548 | . 2 ⊢ (∃*𝑥 𝑥 ∈ 𝐴 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ ⊤)) |
| 4 | df-rmo 3380 | . 2 ⊢ (∃*𝑥 ∈ 𝐴 ⊤ ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ ⊤)) | |
| 5 | 3, 4 | bitr4i 278 | 1 ⊢ (∃*𝑥 𝑥 ∈ 𝐴 ↔ ∃*𝑥 ∈ 𝐴 ⊤) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 ⊤wtru 1541 ∈ wcel 2108 ∃*wmo 2538 ∃*wrmo 3379 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-mo 2540 df-rmo 3380 |
| This theorem is referenced by: reutruALT 48725 mosn 48732 |
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