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Mirrors > Home > MPE Home > Th. List > Mathboxes > reutru | Structured version Visualization version GIF version |
Description: Two ways of expressing "exactly one" element. (Contributed by Zhi Wang, 23-Sep-2024.) |
Ref | Expression |
---|---|
reutru | ⊢ (∃!𝑥 𝑥 ∈ 𝐴 ↔ ∃!𝑥 ∈ 𝐴 ⊤) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tru 1546 | . . . 4 ⊢ ⊤ | |
2 | 1 | biantru 533 | . . 3 ⊢ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∧ ⊤)) |
3 | 2 | eubii 2587 | . 2 ⊢ (∃!𝑥 𝑥 ∈ 𝐴 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ ⊤)) |
4 | df-reu 3061 | . 2 ⊢ (∃!𝑥 ∈ 𝐴 ⊤ ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ ⊤)) | |
5 | 3, 4 | bitr4i 281 | 1 ⊢ (∃!𝑥 𝑥 ∈ 𝐴 ↔ ∃!𝑥 ∈ 𝐴 ⊤) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 ⊤wtru 1543 ∈ wcel 2114 ∃!weu 2570 ∃!wreu 3056 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1545 df-ex 1787 df-mo 2541 df-eu 2571 df-reu 3061 |
This theorem is referenced by: (None) |
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