Theorem bj-denotes 34183

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 2040, and eqeq1 2825, requires the core axioms and { ax-9 2120, ax-ext 2793, df-cleq 2814 } whereas this proof requires the core axioms and { ax-8 2112, df-clab 2800, df-clel 2893 }.

Theorem bj-issetwt 34184 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 2800, df-clel 2893 } (whereas with the shorter proof from cbvexvw 2040 and eqeq1 2825 it would require { ax-8 2112, ax-9 2120, ax-ext 2793, df-clab 2800, df-cleq 2814, df-clel 2893 }). That every class is equal to a class abstraction is proved by abid1 2956, which requires { ax-8 2112, ax-9 2120, ax-ext 2793, df-clab 2800, df-cleq 2814, df-clel 2893 }.

Note that there is no disjoint variable condition on 𝑥, 𝑦 but the theorem does not depend on ax-13 2386. Actually, the proof depends only on the logical axioms ax-1 6 through ax-7 2011 and sp 2178.

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 2793 and df-cleq 2814 (e.g., eqid 2821 and eqeq1 2825). In particular, one cannot even prove ⊢ ∃𝑥𝑥 = 𝐴 ⇒ ⊢ 𝐴 = 𝐴 without ax-ext 2793 and df-cleq 2814.

(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		id 22	. . . . . 6 ⊢ (𝑧 = 𝑧 → 𝑧 = 𝑧)
2	1	vexw 2805	. . . . 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 1844	. . 3 ⊢ (∃𝑥 𝑥 = 𝐴 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ {𝑧 ∣ (𝑧 = 𝑧 → 𝑧 = 𝑧)}))
5		dfclel 2894	. . 3 ⊢ (𝐴 ∈ {𝑧 ∣ (𝑧 = 𝑧 → 𝑧 = 𝑧)} ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ {𝑧 ∣ (𝑧 = 𝑧 → 𝑧 = 𝑧)}))
6		dfclel 2894	. . 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 2805	. . . . 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 1844	. 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)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1533  ∃wex 1776   ∈ wcel 2110  {cab 2799

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1777  df-sb 2066  df-clab 2800  df-clel 2893

This theorem is referenced by:  bj-issetwt  34184  bj-elisset  34187  bj-vtoclg1f1  34228