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Mirrors > Home > ILE Home > Th. List > mosn | GIF version |
Description: A singleton has at most one element. This works whether 𝐴 is a proper class or not, and in that sense can be seen as encompassing both snmg 3636 and snprc 3583. (Contributed by Jim Kingdon, 30-Aug-2018.) |
Ref | Expression |
---|---|
mosn | ⊢ ∃*𝑥 𝑥 ∈ {𝐴} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | moeq 2854 | . 2 ⊢ ∃*𝑥 𝑥 = 𝐴 | |
2 | velsn 3539 | . . 3 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴) | |
3 | 2 | mobii 2034 | . 2 ⊢ (∃*𝑥 𝑥 ∈ {𝐴} ↔ ∃*𝑥 𝑥 = 𝐴) |
4 | 1, 3 | mpbir 145 | 1 ⊢ ∃*𝑥 𝑥 ∈ {𝐴} |
Colors of variables: wff set class |
Syntax hints: = wceq 1331 ∈ wcel 1480 ∃*wmo 1998 {csn 3522 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-v 2683 df-sn 3528 |
This theorem is referenced by: (None) |
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