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Mirrors > Home > ILE Home > Th. List > cvjust | GIF version |
Description: Every set is a class. Proposition 4.9 of [TakeutiZaring] p. 13. This theorem shows that a setvar variable can be expressed as a class abstraction. This provides a motivation for the class syntax construction cv 1315, which allows us to substitute a setvar variable for a class variable. See also cab 2103 and df-clab 2104. Note that this is not a rigorous justification, because cv 1315 is used as part of the proof of this theorem, but a careful argument can be made outside of the formalism of Metamath, for example as is done in Chapter 4 of Takeuti and Zaring. See also the discussion under the definition of class in [Jech] p. 4 showing that "Every set can be considered to be a class." (Contributed by NM, 7-Nov-2006.) |
Ref | Expression |
---|---|
cvjust | ⊢ 𝑥 = {𝑦 ∣ 𝑦 ∈ 𝑥} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfcleq 2111 | . 2 ⊢ (𝑥 = {𝑦 ∣ 𝑦 ∈ 𝑥} ↔ ∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ {𝑦 ∣ 𝑦 ∈ 𝑥})) | |
2 | df-clab 2104 | . . 3 ⊢ (𝑧 ∈ {𝑦 ∣ 𝑦 ∈ 𝑥} ↔ [𝑧 / 𝑦]𝑦 ∈ 𝑥) | |
3 | elsb3 1929 | . . 3 ⊢ ([𝑧 / 𝑦]𝑦 ∈ 𝑥 ↔ 𝑧 ∈ 𝑥) | |
4 | 2, 3 | bitr2i 184 | . 2 ⊢ (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ {𝑦 ∣ 𝑦 ∈ 𝑥}) |
5 | 1, 4 | mpgbir 1414 | 1 ⊢ 𝑥 = {𝑦 ∣ 𝑦 ∈ 𝑥} |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 = wceq 1316 ∈ wcel 1465 [wsb 1720 {cab 2103 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 |
This theorem is referenced by: (None) |
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