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Mirrors > Home > ILE Home > Th. List > clabel | GIF version |
Description: Membership of a class abstraction in another class. (Contributed by NM, 17-Jan-2006.) |
Ref | Expression |
---|---|
clabel | ⊢ ({𝑥 ∣ 𝜑} ∈ 𝐴 ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ ∀𝑥(𝑥 ∈ 𝑦 ↔ 𝜑))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-clel 2133 | . 2 ⊢ ({𝑥 ∣ 𝜑} ∈ 𝐴 ↔ ∃𝑦(𝑦 = {𝑥 ∣ 𝜑} ∧ 𝑦 ∈ 𝐴)) | |
2 | abeq2 2246 | . . . 4 ⊢ (𝑦 = {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝑥 ∈ 𝑦 ↔ 𝜑)) | |
3 | 2 | anbi2ci 454 | . . 3 ⊢ ((𝑦 = {𝑥 ∣ 𝜑} ∧ 𝑦 ∈ 𝐴) ↔ (𝑦 ∈ 𝐴 ∧ ∀𝑥(𝑥 ∈ 𝑦 ↔ 𝜑))) |
4 | 3 | exbii 1584 | . 2 ⊢ (∃𝑦(𝑦 = {𝑥 ∣ 𝜑} ∧ 𝑦 ∈ 𝐴) ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ ∀𝑥(𝑥 ∈ 𝑦 ↔ 𝜑))) |
5 | 1, 4 | bitri 183 | 1 ⊢ ({𝑥 ∣ 𝜑} ∈ 𝐴 ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ ∀𝑥(𝑥 ∈ 𝑦 ↔ 𝜑))) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 ∀wal 1329 = wceq 1331 ∃wex 1468 ∈ wcel 1480 {cab 2123 |
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-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-11 1484 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 |
This theorem is referenced by: frecabcl 6289 |
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