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Mirrors > Home > ILE Home > Th. List > ssab2 | GIF version |
Description: Subclass relation for the restriction of a class abstraction. (Contributed by NM, 31-Mar-1995.) |
Ref | Expression |
---|---|
ssab2 | ⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ⊆ 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 107 | . 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 ∈ 𝐴) | |
2 | 1 | abssi 3070 | 1 ⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ⊆ 𝐴 |
Colors of variables: wff set class |
Syntax hints: ∧ wa 102 ∈ wcel 1434 {cab 2068 ⊆ wss 2974 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2064 |
This theorem depends on definitions: df-bi 115 df-nf 1391 df-sb 1687 df-clab 2069 df-cleq 2075 df-clel 2078 df-nfc 2209 df-in 2980 df-ss 2987 |
This theorem is referenced by: ssrab2 3080 zfausab 3928 exss 3990 dmopabss 4575 fabexg 5108 |
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