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Mirrors > Home > NFE Home > Th. List > a9e | GIF version |
Description: At least one individual exists. This is not a theorem of free logic, which is sound in empty domains. For such a logic, we would add this theorem as an axiom of set theory (Axiom 0 of [Kunen] p. 10). In the system consisting of ax-5 1557 through ax-14 1714 and ax-17 1616, all axioms other than ax9 1949 are believed to be theorems of free logic, although the system without ax9 1949 is probably not complete in free logic. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
a9e | ⊢ ∃x x = y |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax9 1949 | . 2 ⊢ ¬ ∀x ¬ x = y | |
2 | df-ex 1542 | . 2 ⊢ (∃x x = y ↔ ¬ ∀x ¬ x = y) | |
3 | 1, 2 | mpbir 200 | 1 ⊢ ∃x x = y |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∀wal 1540 ∃wex 1541 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 |
This theorem depends on definitions: df-bi 177 df-an 360 df-tru 1319 df-ex 1542 df-nf 1545 |
This theorem is referenced by: equs4 1959 equvini 1987 ax11vALT 2097 axi9 2330 dmi 4919 |
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