Theorem isset 3180

Description: Two ways to say "𝐴 is a set": A class 𝐴 is a member of the universal class V (see df-v 3175) 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 6851. Note the when 𝐴 is not a set, it is called a proper class. In some theorems, such as uniexg 6853, 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 2606 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 2606	. 2 ⊢ (𝐴 ∈ V ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ V))
2		vex 3176	. . . 4 ⊢ 𝑥 ∈ V
3	2	biantru 525	. . 3 ⊢ (𝑥 = 𝐴 ↔ (𝑥 = 𝐴 ∧ 𝑥 ∈ V))
4	3	exbii 1764	. 2 ⊢ (∃𝑥 𝑥 = 𝐴 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ V))
5	1, 4	bitr4i 266	1 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)

Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  Vcvv 3173

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-12 2034  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-an 385  df-tru 1478  df-ex 1696  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-v 3175

This theorem is referenced by:  issetf  3181  isseti  3182  issetri  3183  elex  3185  elisset  3188  vtoclg1f  3238  eueq  3345  moeq  3349  ru  3401  sbc5  3427  snprc  4197  vprc  4724  vnex  4726  eusvnfb  4788  reusv2lem3  4797  iotaex  5785  funimaexg  5889  fvmptdf  6204  fvmptdv2  6206  ovmpt2df  6690  rankf  8540  isssc  16303  snelsingles  31199  bj-snglex  32154  bj-nul  32209  dissneqlem  32363  iotaexeu  37641  elnev  37661  ax6e2nd  37795  ax6e2ndVD  38166  ax6e2ndALT  38188  upbdrech  38460  itgsubsticclem  38867