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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 2766 | . . 3 ⊢ Ⅎ𝑥{𝑥 ∣ 𝑥 ∈ 𝐴} | |
2 | abid2f.1 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
3 | 1, 2 | cleqf 2790 | . 2 ⊢ ({𝑥 ∣ 𝑥 ∈ 𝐴} = 𝐴 ↔ ∀𝑥(𝑥 ∈ {𝑥 ∣ 𝑥 ∈ 𝐴} ↔ 𝑥 ∈ 𝐴)) |
4 | abid 2610 | . 2 ⊢ (𝑥 ∈ {𝑥 ∣ 𝑥 ∈ 𝐴} ↔ 𝑥 ∈ 𝐴) | |
5 | 3, 4 | mpgbir 1726 | 1 ⊢ {𝑥 ∣ 𝑥 ∈ 𝐴} = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 = wceq 1483 ∈ wcel 1990 {cab 2608 Ⅎwnfc 2751 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 |
This theorem is referenced by: mptctf 29495 rabexgf 39183 ssabf 39280 abssf 39295 |
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