Theorem bj-denotesALTV 36831

Description: Moved to main as iseqsetv-clel 2823 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 2744, requires the core axioms and { ax-9 2118, ax-ext 2711, df-cleq 2732 } whereas this proof requires the core axioms and { ax-8 2110, df-clab 2718, df-clel 2819 }.

Theorem bj-issetwt 36834 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 2718, df-clel 2819 } (whereas with the shorter proof from cbvexvw 2036 and eqeq1 2744 it would require { ax-8 2110, ax-9 2118, ax-ext 2711, df-clab 2718, df-cleq 2732, df-clel 2819 }). That every class is equal to a class abstraction is proved by abid1 2881, which requires { ax-8 2110, ax-9 2118, ax-ext 2711, df-clab 2718, df-cleq 2732, df-clel 2819 }.

Note that there is no disjoint variable condition on 𝑥, 𝑦 but the theorem does not depend on ax-13 2380. Actually, the proof depends only on the logical axioms ax-1 6 through ax-7 2007 and sp 2184.

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 2711 and df-cleq 2732 (e.g., eqid 2740 and eqeq1 2744). In particular, one cannot even prove ⊢ ∃𝑥𝑥 = 𝐴 ⇒ ⊢ 𝐴 = 𝐴 without ax-ext 2711 and df-cleq 2732.

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

Assertion

Ref	Expression
bj-denotesALTV	⊢ (∃𝑥 𝑥 = 𝐴 ↔ ∃𝑦 𝑦 = 𝐴)

Distinct variable groups:   𝑥,𝐴   𝑦,𝐴

Proof of Theorem bj-denotesALTV

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bj-denoteslem 36830	. 2 ⊢ (∃𝑥 𝑥 = 𝐴 ↔ 𝐴 ∈ {𝑧 ∣ ⊤})
2		bj-denoteslem 36830	. 2 ⊢ (∃𝑦 𝑦 = 𝐴 ↔ 𝐴 ∈ {𝑧 ∣ ⊤})
3	1, 2	bitr4i 278	1 ⊢ (∃𝑥 𝑥 = 𝐴 ↔ ∃𝑦 𝑦 = 𝐴)

Syntax hints:   ↔ wb 206   = wceq 1537  ⊤wtru 1538  ∃wex 1777   ∈ wcel 2108  {cab 2717

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-clel 2819

This theorem is referenced by: (None)