Theorem isset 3070

Description: Two ways to say "𝐴 is a set": A class 𝐴 is a member of the universal class V (see df-v 3068) if and only if the class 𝐴 exists (i.e. there exists some set 𝑥 equal to class 𝐴). Theorem 6.9 of [Quine] p. 43. Notational convention: We will use the notational device "𝐴 ∈ V " to mean "𝐴 is a set" very frequently, for example in uniex 6663. Note the when 𝐴 is not a set, it is called a proper class. In some theorems, such as uniexg 6665, in order to shorten certain proofs we use the more general antecedent 𝐴 ∈ 𝑉 instead of 𝐴 ∈ V to mean "𝐴 is a set."

Note that a constant is implicitly considered distinct from all variables. This is why V is not included in the distinct variable list, even though df-clel 2501 requires that the expression substituted for 𝐵 not contain 𝑥. (Also, the Metamath spec does not allow constants in the distinct variable list.) (Contributed by NM, 26-May-1993.)

Assertion

Ref	Expression
isset	⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)

Distinct variable group:   𝑥,𝐴

Proof of Theorem isset

Step	Hyp	Ref	Expression
1		df-clel 2501	. 2 ⊢ (𝐴 ∈ V ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ V))
2		vex 3069	. . . 4 ⊢ 𝑥 ∈ V
3	2	biantru 519	. . 3 ⊢ (𝑥 = 𝐴 ↔ (𝑥 = 𝐴 ∧ 𝑥 ∈ V))
4	3	exbii 1749	. 2 ⊢ (∃𝑥 𝑥 = 𝐴 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ V))
5	1, 4	bitr4i 262	1 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)

Syntax hints:   ↔ wb 191   ∧ wa 378   = wceq 1468  ∃wex 1692   ∈ wcel 1937  Vcvv 3066

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1698  ax-4 1711  ax-5 1789  ax-6 1836  ax-7 1883  ax-12 1983  ax-ext 2485

This theorem depends on definitions:  df-bi 192  df-an 380  df-tru 1471  df-ex 1693  df-sb 1829  df-clab 2492  df-cleq 2498  df-clel 2501  df-v 3068

This theorem is referenced by:  issetf  3071  isseti  3072  issetri  3073  elex  3075  elisset  3078  vtoclg1f  3127  eueq  3234  moeq  3238  ru  3290  sbc5  3316  snprc  4064  vprc  4574  vnex  4576  eusvnfb  4638  reusv2lem3  4645  iotaex  5614  funimaexg  5715  fvmptdf  6028  fvmptdv2  6030  ovmpt2df  6503  rankf  8350  isssc  15891  snelsingles  30840  bj-snglex  31753  bj-nul  31808  dissneqlem  31963  iotaexeu  37126  elnev  37146  ax6e2nd  37281  ax6e2ndVD  37653  ax6e2ndALT  37675  upbdrech  37900  itgsubsticclem  38271