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Mirrors > Home > ILE Home > Th. List > elssabg | GIF version |
Description: Membership in a class abstraction involving a subset. Unlike elabg 2688, 𝐴 does not have to be a set. (Contributed by NM, 29-Aug-2006.) |
Ref | Expression |
---|---|
elssabg.1 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
elssabg | ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ {𝑥 ∣ (𝑥 ⊆ 𝐵 ∧ 𝜑)} ↔ (𝐴 ⊆ 𝐵 ∧ 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssexg 3896 | . . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉) → 𝐴 ∈ V) | |
2 | 1 | expcom 109 | . . 3 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ⊆ 𝐵 → 𝐴 ∈ V)) |
3 | 2 | adantrd 264 | . 2 ⊢ (𝐵 ∈ 𝑉 → ((𝐴 ⊆ 𝐵 ∧ 𝜓) → 𝐴 ∈ V)) |
4 | sseq1 2966 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ 𝐵 ↔ 𝐴 ⊆ 𝐵)) | |
5 | elssabg.1 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
6 | 4, 5 | anbi12d 442 | . . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 ⊆ 𝐵 ∧ 𝜑) ↔ (𝐴 ⊆ 𝐵 ∧ 𝜓))) |
7 | 6 | elab3g 2693 | . 2 ⊢ (((𝐴 ⊆ 𝐵 ∧ 𝜓) → 𝐴 ∈ V) → (𝐴 ∈ {𝑥 ∣ (𝑥 ⊆ 𝐵 ∧ 𝜑)} ↔ (𝐴 ⊆ 𝐵 ∧ 𝜓))) |
8 | 3, 7 | syl 14 | 1 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ {𝑥 ∣ (𝑥 ⊆ 𝐵 ∧ 𝜑)} ↔ (𝐴 ⊆ 𝐵 ∧ 𝜓))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 = wceq 1243 ∈ wcel 1393 {cab 2026 Vcvv 2557 ⊆ wss 2917 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-v 2559 df-in 2924 df-ss 2931 |
This theorem is referenced by: (None) |
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