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Mirrors > Home > ILE Home > Th. List > eumo | GIF version |
Description: Existential uniqueness implies "at most one." (Contributed by NM, 23-Mar-1995.) (Proof rewritten by Jim Kingdon, 27-May-2018.) |
Ref | Expression |
---|---|
eumo | ⊢ (∃!𝑥𝜑 → ∃*𝑥𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-1 6 | . 2 ⊢ (∃!𝑥𝜑 → (∃𝑥𝜑 → ∃!𝑥𝜑)) | |
2 | df-mo 2004 | . 2 ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑)) | |
3 | 1, 2 | sylibr 133 | 1 ⊢ (∃!𝑥𝜑 → ∃*𝑥𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∃wex 1469 ∃!weu 2000 ∃*wmo 2001 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 |
This theorem depends on definitions: df-bi 116 df-mo 2004 |
This theorem is referenced by: eumoi 2033 eu5 2047 euimmo 2067 moaneu 2076 eupick 2079 2eumo 2088 moeq3dc 2864 nfunsn 5463 fnoprabg 5880 uptx 12482 txcn 12483 |
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