Theorem isset 3488

Description: Two ways to express that "𝐴 is a set": A class 𝐴 is a member of the universal class V (see df-v 3477) 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.

A class 𝐴 which is not a set is called a proper class.

Conventions: We will often use the expression "𝐴 ∈ V " to mean "𝐴 is a set", for example in uniex 7731. To make some theorems more readily applicable, we will also use the more general expression 𝐴 ∈ 𝑉 instead of 𝐴 ∈ V to mean "𝐴 is a set", typically in an antecedent, or in a hypothesis for theorems in deduction form (see for instance uniexg 7730 compared with uniex 7731). That this is more general is seen either by substitution (when the variable 𝑉 has no other occurrences), or by elex 3493. (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		vex 3479	. 2 ⊢ 𝑥 ∈ V
2	1	issetlem 2814	1 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)

Syntax hints:   ↔ wb 205   = wceq 1542  ∃wex 1782   ∈ wcel 2107  Vcvv 3475

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477

This theorem is referenced by:  issetf  3489  issetri  3491  elex  3493  eueq  3705  ru  3777  sbc5ALT  3807  snprc  4722  snssb  4787  vprc  5316  eusvnfb  5392  reusv2lem3  5399  iotaexOLD  6524  funimaexgOLD  6636  fvmptd3f  7014  fvmptdv2  7017  ovmpodf  7564  rankf  9789  fnpr2ob  17504  isssc  17767  lrrecfr  27427  snelsingles  34894  bj-snglex  35854  bj-abex  35911  bj-clex  35912  bj-nul  35937  dissneqlem  36221  wl-issetft  36444  snen1g  42275  rr-spce  42956  iotaexeu  43177  elnev  43197  ax6e2nd  43319  ax6e2ndVD  43669  ax6e2ndALT  43691  upbdrech  44015  itgsubsticclem  44691