Mathbox for BJ |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-denotes | Structured version Visualization version GIF version |
Description: This would be the
justification theorem for the definition of the unary
predicate "E!" by ⊢ ( E! 𝐴 ↔ ∃𝑥𝑥 = 𝐴) which could be
interpreted as "𝐴 exists" (as a set) or
"𝐴 denotes" (in the
sense of free logic).
A shorter proof using bitri 278 (to add an intermediate proposition ∃𝑧𝑧 = 𝐴 with a fresh 𝑧), cbvexvw 2044, and eqeq1 2802, requires the core axioms and { ax-9 2121, ax-ext 2770, df-cleq 2791 } whereas this proof requires the core axioms and { ax-8 2113, df-clab 2777, df-clel 2870 }. Theorem bj-issetwt 34313 proves that "existing" is equivalent to being a member of a class abstraction. It also requires, with the present proof, { ax-8 2113, df-clab 2777, df-clel 2870 } (whereas with the shorter proof from cbvexvw 2044 and eqeq1 2802 it would require { ax-8 2113, ax-9 2121, ax-ext 2770, df-clab 2777, df-cleq 2791, df-clel 2870 }). That every class is equal to a class abstraction is proved by abid1 2931, which requires { ax-8 2113, ax-9 2121, ax-ext 2770, df-clab 2777, df-cleq 2791, df-clel 2870 }. Note that there is no disjoint variable condition on 𝑥, 𝑦 but the theorem does not depend on ax-13 2379. Actually, the proof depends only on the logical axioms ax-1 6 through ax-7 2015 and sp 2180. The symbol "E!" was chosen to be reminiscent of the analogous predicate in (inclusive or non-inclusive) free logic, which deals with the possibility of nonexistent objects. This analogy should not be taken too far, since here there are no equality axioms for classes: these are derived from ax-ext 2770 and df-cleq 2791 (e.g., eqid 2798 and eqeq1 2802). In particular, one cannot even prove ⊢ ∃𝑥𝑥 = 𝐴 ⇒ ⊢ 𝐴 = 𝐴 without ax-ext 2770 and df-cleq 2791. (Contributed by BJ, 29-Apr-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-denotes | ⊢ (∃𝑥 𝑥 = 𝐴 ↔ ∃𝑦 𝑦 = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-denoteslem 34309 | . 2 ⊢ (∃𝑥 𝑥 = 𝐴 ↔ 𝐴 ∈ {𝑧 ∣ ⊤}) | |
2 | bj-denoteslem 34309 | . 2 ⊢ (∃𝑦 𝑦 = 𝐴 ↔ 𝐴 ∈ {𝑧 ∣ ⊤}) | |
3 | 1, 2 | bitr4i 281 | 1 ⊢ (∃𝑥 𝑥 = 𝐴 ↔ ∃𝑦 𝑦 = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 = wceq 1538 ⊤wtru 1539 ∃wex 1781 ∈ wcel 2111 {cab 2776 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1541 df-ex 1782 df-sb 2070 df-clab 2777 df-clel 2870 |
This theorem is referenced by: bj-issettru 34311 bj-issetwt 34313 bj-elisset 34316 bj-vtoclg1f1 34357 |
Copyright terms: Public domain | W3C validator |