Mathbox for Wolf Lammen |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > df-wl-rmo | Structured version Visualization version GIF version |
Description: Restrict "at most
one" to a given class 𝐴. This version does not
interpret elementhood verbatim like ∃*𝑥 ∈ 𝐴𝜑 does. Assuming a
real elementhood can lead to awkward consequences should the class 𝐴
depend on 𝑥. Instead we base the definition on
df-wl-ral 34881, where
this is already ruled out.
This definition lets 𝑥 appear as a formal parameter with no connection to 𝐴, but occurrences in 𝜑 are fully honored. Alternate definitions are given in wl-dfrmosb 34898 and, if 𝑥 is not free in 𝐴, wl-dfrmov 34899 and wl-dfrmof 34900. (Contributed by NM, 30-Aug-1993.) Isolate x from A. (Revised by Wolf Lammen, 26-May-2023.) |
Ref | Expression |
---|---|
df-wl-rmo | ⊢ (∃*(𝑥 : 𝐴)𝜑 ↔ ∃𝑦∀(𝑥 : 𝐴)(𝜑 → 𝑥 = 𝑦)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wph | . . 3 wff 𝜑 | |
2 | vx | . . 3 setvar 𝑥 | |
3 | cA | . . 3 class 𝐴 | |
4 | 1, 2, 3 | wl-rmo 34878 | . 2 wff ∃*(𝑥 : 𝐴)𝜑 |
5 | vy | . . . . . 6 setvar 𝑦 | |
6 | 2, 5 | weq 1963 | . . . . 5 wff 𝑥 = 𝑦 |
7 | 1, 6 | wi 4 | . . . 4 wff (𝜑 → 𝑥 = 𝑦) |
8 | 7, 2, 3 | wl-ral 34876 | . . 3 wff ∀(𝑥 : 𝐴)(𝜑 → 𝑥 = 𝑦) |
9 | 8, 5 | wex 1779 | . 2 wff ∃𝑦∀(𝑥 : 𝐴)(𝜑 → 𝑥 = 𝑦) |
10 | 4, 9 | wb 208 | 1 wff (∃*(𝑥 : 𝐴)𝜑 ↔ ∃𝑦∀(𝑥 : 𝐴)(𝜑 → 𝑥 = 𝑦)) |
Colors of variables: wff setvar class |
This definition is referenced by: wl-dfrmosb 34898 wl-dfrmov 34899 wl-dfrmof 34900 |
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