Theorem isset 2810

Description: Two ways to say "𝐴 is a set": A class 𝐴 is a member of the universal class V (see df-v 2805) 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 4540. Note the when 𝐴 is not a set, it is called a proper class. In some theorems, such as uniexg 4542, 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 2227 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 2227	. 2 ⊢ (𝐴 ∈ V ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ V))
2		vex 2806	. . . 4 ⊢ 𝑥 ∈ V
3	2	biantru 302	. . 3 ⊢ (𝑥 = 𝐴 ↔ (𝑥 = 𝐴 ∧ 𝑥 ∈ V))
4	3	exbii 1654	. 2 ⊢ (∃𝑥 𝑥 = 𝐴 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ V))
5	1, 4	bitr4i 187	1 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)

Syntax hints:   ∧ wa 104   ↔ wb 105   = wceq 1398  ∃wex 1541   ∈ wcel 2202  Vcvv 2803

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1496  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-v 2805

This theorem is referenced by:  issetf  2811  isseti  2812  issetri  2813  elex  2815  elisset  2818  vtoclg1f  2864  ceqex  2934  eueq  2978  moeq  2982  mosubt  2984  ru  3031  sbc5  3056  snprc  3738  rabsnif  3742  snmb  3797  snssb  3811  vprc  4226  opelopabsb  4360  eusvnfb  4557  elrelimasn  5109  euiotaex  5310  fvmptdf  5743  fvmptdv2  5745  fmptco  5821  brabvv  6077  ovmpodf  6163  ovi3  6169  tfrlemibxssdm  6536  tfr1onlembxssdm  6552  tfrcllembxssdm  6565  ecexr  6750  snexxph  7192  fnpr2ob  13486  bj-vprc  16595  bj-vnex  16597  bj-2inf  16637