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Mirrors > Home > MPE Home > Th. List > reurmo | Structured version Visualization version GIF version |
Description: Restricted existential uniqueness implies restricted "at most one." (Contributed by NM, 16-Jun-2017.) |
Ref | Expression |
---|---|
reurmo | ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → ∃*𝑥 ∈ 𝐴 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reu5 3136 | . 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃*𝑥 ∈ 𝐴 𝜑)) | |
2 | 1 | simprbi 479 | 1 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → ∃*𝑥 ∈ 𝐴 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∃wrex 2897 ∃!wreu 2898 ∃*wrmo 2899 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 |
This theorem depends on definitions: df-bi 196 df-an 385 df-ex 1696 df-eu 2462 df-mo 2463 df-rex 2902 df-reu 2903 df-rmo 2904 |
This theorem is referenced by: reuxfrd 4819 enqeq 9635 eqsqrtd 13955 efgred2 17989 0frgp 18015 frgpnabllem2 18100 frgpcyg 19741 lmieu 25476 reuxfr4d 28714 poimirlem25 32604 poimirlem26 32605 reuimrmo 39827 2reurmo 39831 2rexreu 39834 2reu2 39836 |
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