Theorem bj-denotes 36055

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 274 (to add an intermediate proposition ∃𝑧𝑧 = 𝐴 with a fresh 𝑧), cbvexvw 2039, and eqeq1 2735, requires the core axioms and { ax-9 2115, ax-ext 2702, df-cleq 2723 } whereas this proof requires the core axioms and { ax-8 2107, df-clab 2709, df-clel 2809 }.

Theorem bj-issetwt 36058 proves that "existing" is equivalent to being a member of a class abstraction. It also requires, with the present proof, { ax-8 2107, df-clab 2709, df-clel 2809 } (whereas with the shorter proof from cbvexvw 2039 and eqeq1 2735 it would require { ax-8 2107, ax-9 2115, ax-ext 2702, df-clab 2709, df-cleq 2723, df-clel 2809 }). That every class is equal to a class abstraction is proved by abid1 2869, which requires { ax-8 2107, ax-9 2115, ax-ext 2702, df-clab 2709, df-cleq 2723, df-clel 2809 }.

Note that there is no disjoint variable condition on 𝑥, 𝑦 but the theorem does not depend on ax-13 2370. Actually, the proof depends only on the logical axioms ax-1 6 through ax-7 2010 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 2702 and df-cleq 2723 (e.g., eqid 2731 and eqeq1 2735). In particular, one cannot even prove ⊢ ∃𝑥𝑥 = 𝐴 ⇒ ⊢ 𝐴 = 𝐴 without ax-ext 2702 and df-cleq 2723.

(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.

Step	Hyp	Ref	Expression
1		bj-denoteslem 36054	. 2 ⊢ (∃𝑥 𝑥 = 𝐴 ↔ 𝐴 ∈ {𝑧 ∣ ⊤})
2		bj-denoteslem 36054	. 2 ⊢ (∃𝑦 𝑦 = 𝐴 ↔ 𝐴 ∈ {𝑧 ∣ ⊤})
3	1, 2	bitr4i 277	1 ⊢ (∃𝑥 𝑥 = 𝐴 ↔ ∃𝑦 𝑦 = 𝐴)

Syntax hints:   ↔ wb 205   = wceq 1540  ⊤wtru 1541  ∃wex 1780   ∈ wcel 2105  {cab 2708

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2709  df-clel 2809

This theorem is referenced by:  bj-issettru  36056  bj-issetwt  36058  bj-elissetALT  36060  bj-vtoclg1f1  36101