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Mirrors > Home > MPE Home > Th. List > cvjust | Structured version Visualization version 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 1537, which allows us to substitute a setvar variable for a class variable. See also cab 2776 and df-clab 2777. Note that this is not a rigorous justification, because cv 1537 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." See abid1 2931 for the version of cvjust 2793 extended to classes. (Contributed by NM, 7-Nov-2006.) Avoid ax-13 2379. (Revised by Wolf Lammen, 4-May-2023.) |
Ref | Expression |
---|---|
cvjust | ⊢ 𝑥 = {𝑦 ∣ 𝑦 ∈ 𝑥} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfcleq 2792 | . 2 ⊢ (𝑥 = {𝑦 ∣ 𝑦 ∈ 𝑥} ↔ ∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ {𝑦 ∣ 𝑦 ∈ 𝑥})) | |
2 | df-clab 2777 | . . 3 ⊢ (𝑧 ∈ {𝑦 ∣ 𝑦 ∈ 𝑥} ↔ [𝑧 / 𝑦]𝑦 ∈ 𝑥) | |
3 | elsb3 2119 | . . 3 ⊢ ([𝑧 / 𝑦]𝑦 ∈ 𝑥 ↔ 𝑧 ∈ 𝑥) | |
4 | 2, 3 | bitr2i 279 | . 2 ⊢ (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ {𝑦 ∣ 𝑦 ∈ 𝑥}) |
5 | 1, 4 | mpgbir 1801 | 1 ⊢ 𝑥 = {𝑦 ∣ 𝑦 ∈ 𝑥} |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 = wceq 1538 [wsb 2069 ∈ wcel 2111 {cab 2776 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 |
This theorem is referenced by: cnambfre 35105 |
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