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| Mirrors > Home > MPE Home > Th. List > isset | Structured version Visualization version GIF version | ||
| Description: Two ways to express that
"𝐴 is a set": A class 𝐴 is a
member
of the universal class V (see df-v 3482)
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 7761. 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 7760 compared with uniex 7761). That this is more general is seen either by substitution (when the variable 𝑉 has no other occurrences), or by elex 3501. (Contributed by NM, 26-May-1993.) |
| Ref | Expression |
|---|---|
| isset | ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | vex 3484 | . 2 ⊢ 𝑥 ∈ V | |
| 2 | 1 | issetlem 2821 | 1 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 = wceq 1540 ∃wex 1779 ∈ wcel 2108 Vcvv 3480 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 |
| This theorem is referenced by: issetft 3496 issetri 3499 elex 3501 elexOLD 3502 eueq 3714 ru 3786 ruOLD 3787 sbc5ALT 3817 sbccomlem 3869 snprc 4717 snssb 4782 vprc 5315 eusvnfb 5393 reusv2lem3 5400 iotaexOLD 6541 funimaexgOLD 6654 fvmptd3f 7031 fvmptdv2 7034 ovmpodf 7589 rankf 9834 fnpr2ob 17603 isssc 17864 lrrecfr 27976 snelsingles 35923 bj-snglex 36974 bj-abex 37031 bj-clex 37032 bj-nul 37057 dissneqlem 37341 wl-issetft 37583 snen1g 43537 rr-spce 44217 iotaexeu 44437 elnev 44457 ax6e2nd 44578 ax6e2ndVD 44928 ax6e2ndALT 44950 upbdrech 45317 itgsubsticclem 45990 |
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