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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-denotesALTV | Structured version Visualization version GIF version | ||
| Description: Moved to main as iseqsetv-clel 2819 and kept for the comments.
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 275 (to add an intermediate proposition ∃𝑧𝑧 = 𝐴 with a fresh 𝑧), cbvexvw 2036, and eqeq1 2740, requires the core axioms and { ax-9 2118, ax-ext 2707, df-cleq 2728 } whereas this proof requires the core axioms and { ax-8 2110, df-clab 2714, df-clel 2815 }. Theorem bj-issetwt 36854 proves that "existing" is equivalent to being a member of a class abstraction. It also requires, with the present proof, { ax-8 2110, df-clab 2714, df-clel 2815 } (whereas with the shorter proof from cbvexvw 2036 and eqeq1 2740 it would require { ax-8 2110, ax-9 2118, ax-ext 2707, df-clab 2714, df-cleq 2728, df-clel 2815 }). That every class is equal to a class abstraction is proved by abid1 2877, which requires { ax-8 2110, ax-9 2118, ax-ext 2707, df-clab 2714, df-cleq 2728, df-clel 2815 }. Note that there is no disjoint variable condition on 𝑥, 𝑦 but the theorem does not depend on ax-13 2376. Actually, the proof depends only on the logical axioms ax-1 6 through ax-7 2007 and sp 2183. 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 2707 and df-cleq 2728 (e.g., eqid 2736 and eqeq1 2740). In particular, one cannot even prove ⊢ ∃𝑥𝑥 = 𝐴 ⇒ ⊢ 𝐴 = 𝐴 without ax-ext 2707 and df-cleq 2728. (Contributed by BJ, 29-Apr-2019.) (Proof modification is discouraged.) |
| Ref | Expression |
|---|---|
| bj-denotesALTV | ⊢ (∃𝑥 𝑥 = 𝐴 ↔ ∃𝑦 𝑦 = 𝐴) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | bj-denoteslem 36850 | . 2 ⊢ (∃𝑥 𝑥 = 𝐴 ↔ 𝐴 ∈ {𝑧 ∣ ⊤}) | |
| 2 | bj-denoteslem 36850 | . 2 ⊢ (∃𝑦 𝑦 = 𝐴 ↔ 𝐴 ∈ {𝑧 ∣ ⊤}) | |
| 3 | 1, 2 | bitr4i 278 | 1 ⊢ (∃𝑥 𝑥 = 𝐴 ↔ ∃𝑦 𝑦 = 𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 = wceq 1540 ⊤wtru 1541 ∃wex 1779 ∈ wcel 2108 {cab 2713 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2714 df-clel 2815 |
| This theorem is referenced by: (None) |
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