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Mirrors > Home > ILE Home > Th. List > abid2f | 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.) |
Ref | Expression |
---|---|
abid2f.1 | ⊢ Ⅎ𝑥𝐴 |
Ref | Expression |
---|---|
abid2f | ⊢ {𝑥 ∣ 𝑥 ∈ 𝐴} = 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | abid2f.1 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
2 | nfab1 2237 | . . . . 5 ⊢ Ⅎ𝑥{𝑥 ∣ 𝑥 ∈ 𝐴} | |
3 | 1, 2 | cleqf 2259 | . . . 4 ⊢ (𝐴 = {𝑥 ∣ 𝑥 ∈ 𝐴} ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑥 ∣ 𝑥 ∈ 𝐴})) |
4 | abid 2083 | . . . . . 6 ⊢ (𝑥 ∈ {𝑥 ∣ 𝑥 ∈ 𝐴} ↔ 𝑥 ∈ 𝐴) | |
5 | 4 | bibi2i 226 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑥 ∣ 𝑥 ∈ 𝐴}) ↔ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴)) |
6 | 5 | albii 1411 | . . . 4 ⊢ (∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑥 ∣ 𝑥 ∈ 𝐴}) ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴)) |
7 | 3, 6 | bitri 183 | . . 3 ⊢ (𝐴 = {𝑥 ∣ 𝑥 ∈ 𝐴} ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴)) |
8 | biid 170 | . . 3 ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴) | |
9 | 7, 8 | mpgbir 1394 | . 2 ⊢ 𝐴 = {𝑥 ∣ 𝑥 ∈ 𝐴} |
10 | 9 | eqcomi 2099 | 1 ⊢ {𝑥 ∣ 𝑥 ∈ 𝐴} = 𝐴 |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 ∀wal 1294 = wceq 1296 ∈ wcel 1445 {cab 2081 Ⅎwnfc 2222 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 |
This theorem depends on definitions: df-bi 116 df-tru 1299 df-nf 1402 df-sb 1700 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 |
This theorem is referenced by: (None) |
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