Theorem bj-denotes 34750

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 2045, and eqeq1 2738, requires the core axioms and { ax-9 2120, ax-ext 2706, df-cleq 2726 } whereas this proof requires the core axioms and { ax-8 2112, df-clab 2713, df-clel 2812 }.

Theorem bj-issetwt 34753 proves that "existing" is equivalent to being a member of a class abstraction. It also requires, with the present proof, { ax-8 2112, df-clab 2713, df-clel 2812 } (whereas with the shorter proof from cbvexvw 2045 and eqeq1 2738 it would require { ax-8 2112, ax-9 2120, ax-ext 2706, df-clab 2713, df-cleq 2726, df-clel 2812 }). That every class is equal to a class abstraction is proved by abid1 2874, which requires { ax-8 2112, ax-9 2120, ax-ext 2706, df-clab 2713, df-cleq 2726, df-clel 2812 }.

Note that there is no disjoint variable condition on 𝑥, 𝑦 but the theorem does not depend on ax-13 2369. Actually, the proof depends only on the logical axioms ax-1 6 through ax-7 2016 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 2706 and df-cleq 2726 (e.g., eqid 2734 and eqeq1 2738). In particular, one cannot even prove ⊢ ∃𝑥𝑥 = 𝐴 ⇒ ⊢ 𝐴 = 𝐴 without ax-ext 2706 and df-cleq 2726.

(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 34749	. 2 ⊢ (∃𝑥 𝑥 = 𝐴 ↔ 𝐴 ∈ {𝑧 ∣ ⊤})
2		bj-denoteslem 34749	. 2 ⊢ (∃𝑦 𝑦 = 𝐴 ↔ 𝐴 ∈ {𝑧 ∣ ⊤})
3	1, 2	bitr4i 281	1 ⊢ (∃𝑥 𝑥 = 𝐴 ↔ ∃𝑦 𝑦 = 𝐴)

Syntax hints:   ↔ wb 209   = wceq 1543  ⊤wtru 1544  ∃wex 1787   ∈ wcel 2110  {cab 2712

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112

This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1546  df-ex 1788  df-sb 2071  df-clab 2713  df-clel 2812

This theorem is referenced by:  bj-issettru  34751  bj-issetwt  34753  bj-elisset  34755  bj-vtoclg1f1  34796