Theorem isset 3408

Description: Two ways to say "𝐴 is a set": A class 𝐴 is a member of the universal class V (see df-v 3399) 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 7230. Note that a class 𝐴 which is not a set is called a proper class. In some theorems, such as uniexg 7232, 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 2773 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 2773	. 2 ⊢ (𝐴 ∈ V ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ V))
2		vex 3400	. . . 4 ⊢ 𝑥 ∈ V
3	2	biantru 525	. . 3 ⊢ (𝑥 = 𝐴 ↔ (𝑥 = 𝐴 ∧ 𝑥 ∈ V))
4	3	exbii 1892	. 2 ⊢ (∃𝑥 𝑥 = 𝐴 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ V))
5	1, 4	bitr4i 270	1 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)

Syntax hints:   ↔ wb 198   ∧ wa 386   = wceq 1601  ∃wex 1823   ∈ wcel 2106  Vcvv 3397

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-9 2115  ax-12 2162  ax-ext 2753

This theorem depends on definitions:  df-bi 199  df-an 387  df-tru 1605  df-ex 1824  df-sb 2012  df-clab 2763  df-cleq 2769  df-clel 2773  df-v 3399

This theorem is referenced by:  issetf  3409  isseti  3410  issetri  3411  elex  3413  elisset  3416  vtoclg1f  3465  eueq  3588  eueqOLD  3589  moeqOLD  3593  ru  3650  sbc5  3676  snprc  4483  vprc  5034  eusvnfb  5105  reusv2lem3  5112  iotaex  6116  funimaexg  6220  fvmptd3f  6556  fvmptdv2  6559  ovmpt2df  7069  rankf  8954  isssc  16865  dmscut  32521  snelsingles  32632  bj-snglex  33547  bj-nul  33604  dissneqlem  33797  iotaexeu  39566  elnev  39586  ax6e2nd  39710  ax6e2ndVD  40069  ax6e2ndALT  40091  upbdrech  40420  itgsubsticclem  41110