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Theorem bj-denotes 34310
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 = 𝐴𝐴 = 𝐴 without ax-ext 2770 and df-cleq 2791.

(Contributed by BJ, 29-Apr-2019.) (Proof modification is discouraged.)

Assertion
Ref Expression
bj-denotes (∃𝑥 𝑥 = 𝐴 ↔ ∃𝑦 𝑦 = 𝐴)
Distinct variable groups:   𝑥,𝐴   𝑦,𝐴

Proof of Theorem bj-denotes
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 bj-denoteslem 34309 . 2 (∃𝑥 𝑥 = 𝐴𝐴 ∈ {𝑧 ∣ ⊤})
2 bj-denoteslem 34309 . 2 (∃𝑦 𝑦 = 𝐴𝐴 ∈ {𝑧 ∣ ⊤})
31, 2bitr4i 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
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