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Mirrors > Home > MPE Home > Th. List > abid2f | Structured version Visualization version GIF version |
Description: A simplification of class abstraction. Theorem 5.2 of [Quine] p. 35. (Contributed by NM, 5-Sep-2011.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 17-Nov-2019.) |
Ref | Expression |
---|---|
abid2f.1 | ⊢ Ⅎ𝑥𝐴 |
Ref | Expression |
---|---|
abid2f | ⊢ {𝑥 ∣ 𝑥 ∈ 𝐴} = 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfab1 2753 | . . 3 ⊢ Ⅎ𝑥{𝑥 ∣ 𝑥 ∈ 𝐴} | |
2 | abid2f.1 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
3 | 1, 2 | cleqf 2776 | . 2 ⊢ ({𝑥 ∣ 𝑥 ∈ 𝐴} = 𝐴 ↔ ∀𝑥(𝑥 ∈ {𝑥 ∣ 𝑥 ∈ 𝐴} ↔ 𝑥 ∈ 𝐴)) |
4 | abid 2598 | . 2 ⊢ (𝑥 ∈ {𝑥 ∣ 𝑥 ∈ 𝐴} ↔ 𝑥 ∈ 𝐴) | |
5 | 3, 4 | mpgbir 1717 | 1 ⊢ {𝑥 ∣ 𝑥 ∈ 𝐴} = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 = wceq 1475 ∈ wcel 1977 {cab 2596 Ⅎwnfc 2738 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 |
This theorem is referenced by: mptctf 28883 rabexgf 38206 ssabf 38308 abssf 38326 |
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