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Mirrors > Home > ILE Home > Th. List > rabss2 | GIF version |
Description: Subclass law for restricted abstraction. (Contributed by NM, 18-Dec-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
Ref | Expression |
---|---|
rabss2 | ⊢ (𝐴 ⊆ 𝐵 → {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ {𝑥 ∈ 𝐵 ∣ 𝜑}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm3.45 561 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵) → ((𝑥 ∈ 𝐴 ∧ 𝜑) → (𝑥 ∈ 𝐵 ∧ 𝜑))) | |
2 | 1 | alimi 1384 | . . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵) → ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → (𝑥 ∈ 𝐵 ∧ 𝜑))) |
3 | dfss2 2988 | . . 3 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) | |
4 | ss2ab 3062 | . . 3 ⊢ ({𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ⊆ {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)} ↔ ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → (𝑥 ∈ 𝐵 ∧ 𝜑))) | |
5 | 2, 3, 4 | 3imtr4i 199 | . 2 ⊢ (𝐴 ⊆ 𝐵 → {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ⊆ {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)}) |
6 | df-rab 2357 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} | |
7 | df-rab 2357 | . 2 ⊢ {𝑥 ∈ 𝐵 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)} | |
8 | 5, 6, 7 | 3sstr4g 3040 | 1 ⊢ (𝐴 ⊆ 𝐵 → {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ {𝑥 ∈ 𝐵 ∣ 𝜑}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ∀wal 1282 ∈ wcel 1433 {cab 2067 {crab 2352 ⊆ wss 2973 |
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 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-rab 2357 df-in 2979 df-ss 2986 |
This theorem is referenced by: sess2 4093 |
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