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Mirrors > Home > ILE Home > Th. List > isset | GIF version |
Description: Two ways to say
"𝐴 is a set": A class 𝐴 is a
member of the
universal class V (see df-v 2614)
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 4228. Note the when 𝐴 is not
a set,
it is called a proper class. In some theorems, such as uniexg 4229, 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 2079 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.) |
Ref | Expression |
---|---|
isset | ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-clel 2079 | . 2 ⊢ (𝐴 ∈ V ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ V)) | |
2 | vex 2615 | . . . 4 ⊢ 𝑥 ∈ V | |
3 | 2 | biantru 296 | . . 3 ⊢ (𝑥 = 𝐴 ↔ (𝑥 = 𝐴 ∧ 𝑥 ∈ V)) |
4 | 3 | exbii 1537 | . 2 ⊢ (∃𝑥 𝑥 = 𝐴 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ V)) |
5 | 1, 4 | bitr4i 185 | 1 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 102 ↔ wb 103 = wceq 1285 ∃wex 1422 ∈ wcel 1434 Vcvv 2612 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-5 1377 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-4 1441 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-ext 2065 |
This theorem depends on definitions: df-bi 115 df-sb 1688 df-clab 2070 df-cleq 2076 df-clel 2079 df-v 2614 |
This theorem is referenced by: issetf 2617 isseti 2618 issetri 2619 elex 2621 elisset 2624 ceqex 2732 eueq 2774 moeq 2778 mosubt 2780 ru 2825 sbc5 2849 snprc 3481 vprc 3936 opelopabsb 4051 eusvnfb 4240 euiotaex 4950 fvmptdf 5335 fvmptdv2 5337 fmptco 5406 brabvv 5630 ovmpt2df 5711 ovi3 5716 tfrlemibxssdm 6024 tfr1onlembxssdm 6040 tfrcllembxssdm 6053 ecexr 6227 snexxph 6583 bj-vprc 11130 bj-vnex 11132 bj-2inf 11176 |
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