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Mirrors > Home > MPE Home > Th. List > rmo5 | Structured version Visualization version GIF version |
Description: Restricted "at most one" in term of uniqueness. (Contributed by NM, 16-Jun-2017.) |
Ref | Expression |
---|---|
rmo5 | ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑥 ∈ 𝐴 𝜑 → ∃!𝑥 ∈ 𝐴 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-mo 2475 | . 2 ⊢ (∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) → ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))) | |
2 | df-rmo 2920 | . 2 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
3 | df-rex 2918 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
4 | df-reu 2919 | . . 3 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
5 | 3, 4 | imbi12i 340 | . 2 ⊢ ((∃𝑥 ∈ 𝐴 𝜑 → ∃!𝑥 ∈ 𝐴 𝜑) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) → ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))) |
6 | 1, 2, 5 | 3bitr4i 292 | 1 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑥 ∈ 𝐴 𝜑 → ∃!𝑥 ∈ 𝐴 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∃wex 1704 ∈ wcel 1990 ∃!weu 2470 ∃*wmo 2471 ∃wrex 2913 ∃!wreu 2914 ∃*wrmo 2915 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 |
This theorem depends on definitions: df-bi 197 df-mo 2475 df-rex 2918 df-reu 2919 df-rmo 2920 |
This theorem is referenced by: nrexrmo 3163 cbvrmo 3170 ddemeas 30299 2reurex 41181 iccpartdisj 41373 |
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