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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 277 (to add an intermediate proposition ∃𝑧𝑧 = 𝐴 with a fresh 𝑧), cbvexvw 2038, and eqeq1 2823, requires the core axioms and { ax-9 2118, ax-ext 2791, df-cleq 2812 } whereas this proof requires the core axioms and { ax-8 2110, df-clab 2798, df-clel 2891 }. Theorem bj-issetwt 34182 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 2798, df-clel 2891 } (whereas with the shorter proof from cbvexvw 2038 and eqeq1 2823 it would require { ax-8 2110, ax-9 2118, ax-ext 2791, df-clab 2798, df-cleq 2812, df-clel 2891 }). That every class is equal to a class abstraction is proved by abid1 2954, which requires { ax-8 2110, ax-9 2118, ax-ext 2791, df-clab 2798, df-cleq 2812, df-clel 2891 }. Note that there is no disjoint variable condition on 𝑥, 𝑦 but the theorem does not depend on ax-13 2384. Actually, the proof depends only on the logical axioms ax-1 6 through ax-7 2009 and sp 2175. 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 2791 and df-cleq 2812 (e.g., eqid 2819 and eqeq1 2823). In particular, one cannot even prove ⊢ ∃𝑥𝑥 = 𝐴 ⇒ ⊢ 𝐴 = 𝐴 without ax-ext 2791 and df-cleq 2812. (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 | id 22 | . . . . . 6 ⊢ (𝑧 = 𝑧 → 𝑧 = 𝑧) | |
2 | 1 | vexw 2803 | . . . . 5 ⊢ 𝑥 ∈ {𝑧 ∣ (𝑧 = 𝑧 → 𝑧 = 𝑧)} |
3 | 2 | biantru 532 | . . . 4 ⊢ (𝑥 = 𝐴 ↔ (𝑥 = 𝐴 ∧ 𝑥 ∈ {𝑧 ∣ (𝑧 = 𝑧 → 𝑧 = 𝑧)})) |
4 | 3 | exbii 1842 | . . 3 ⊢ (∃𝑥 𝑥 = 𝐴 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ {𝑧 ∣ (𝑧 = 𝑧 → 𝑧 = 𝑧)})) |
5 | dfclel 2892 | . . 3 ⊢ (𝐴 ∈ {𝑧 ∣ (𝑧 = 𝑧 → 𝑧 = 𝑧)} ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ {𝑧 ∣ (𝑧 = 𝑧 → 𝑧 = 𝑧)})) | |
6 | dfclel 2892 | . . 3 ⊢ (𝐴 ∈ {𝑧 ∣ (𝑧 = 𝑧 → 𝑧 = 𝑧)} ↔ ∃𝑦(𝑦 = 𝐴 ∧ 𝑦 ∈ {𝑧 ∣ (𝑧 = 𝑧 → 𝑧 = 𝑧)})) | |
7 | 4, 5, 6 | 3bitr2i 301 | . 2 ⊢ (∃𝑥 𝑥 = 𝐴 ↔ ∃𝑦(𝑦 = 𝐴 ∧ 𝑦 ∈ {𝑧 ∣ (𝑧 = 𝑧 → 𝑧 = 𝑧)})) |
8 | 1 | vexw 2803 | . . . . 5 ⊢ 𝑦 ∈ {𝑧 ∣ (𝑧 = 𝑧 → 𝑧 = 𝑧)} |
9 | 8 | biantru 532 | . . . 4 ⊢ (𝑦 = 𝐴 ↔ (𝑦 = 𝐴 ∧ 𝑦 ∈ {𝑧 ∣ (𝑧 = 𝑧 → 𝑧 = 𝑧)})) |
10 | 9 | bicomi 226 | . . 3 ⊢ ((𝑦 = 𝐴 ∧ 𝑦 ∈ {𝑧 ∣ (𝑧 = 𝑧 → 𝑧 = 𝑧)}) ↔ 𝑦 = 𝐴) |
11 | 10 | exbii 1842 | . 2 ⊢ (∃𝑦(𝑦 = 𝐴 ∧ 𝑦 ∈ {𝑧 ∣ (𝑧 = 𝑧 → 𝑧 = 𝑧)}) ↔ ∃𝑦 𝑦 = 𝐴) |
12 | 7, 11 | bitri 277 | 1 ⊢ (∃𝑥 𝑥 = 𝐴 ↔ ∃𝑦 𝑦 = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1531 ∃wex 1774 ∈ wcel 2108 {cab 2797 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1775 df-sb 2064 df-clab 2798 df-clel 2891 |
This theorem is referenced by: bj-issetwt 34182 bj-elisset 34185 bj-vtoclg1f1 34226 |
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