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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 | moeu 2661 | . 2 ⊢ (∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) → ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))) | |
2 | df-rmo 3143 | . 2 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
3 | df-rex 3141 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
4 | df-reu 3142 | . . 3 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
5 | 3, 4 | imbi12i 352 | . 2 ⊢ ((∃𝑥 ∈ 𝐴 𝜑 → ∃!𝑥 ∈ 𝐴 𝜑) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) → ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))) |
6 | 1, 2, 5 | 3bitr4i 304 | 1 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑥 ∈ 𝐴 𝜑 → ∃!𝑥 ∈ 𝐴 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∃wex 1771 ∈ wcel 2105 ∃*wmo 2613 ∃!weu 2646 ∃wrex 3136 ∃!wreu 3137 ∃*wrmo 3138 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 |
This theorem depends on definitions: df-bi 208 df-an 397 df-ex 1772 df-mo 2615 df-eu 2647 df-rex 3141 df-reu 3142 df-rmo 3143 |
This theorem is referenced by: nrexrmo 3433 cbvrmow 3442 cbvrmo 3446 2reurex 3747 rmo0 4316 rmosn 4647 ddemeas 31394 iccpartdisj 43474 |
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