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Mirrors > Home > ILE Home > Th. List > elint | GIF version |
Description: Membership in class intersection. (Contributed by NM, 21-May-1994.) |
Ref | Expression |
---|---|
elint.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
elint | ⊢ (𝐴 ∈ ∩ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝐴 ∈ 𝑥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elint.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | eleq1 2100 | . . . 4 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ 𝑥 ↔ 𝐴 ∈ 𝑥)) | |
3 | 2 | imbi2d 219 | . . 3 ⊢ (𝑦 = 𝐴 → ((𝑥 ∈ 𝐵 → 𝑦 ∈ 𝑥) ↔ (𝑥 ∈ 𝐵 → 𝐴 ∈ 𝑥))) |
4 | 3 | albidv 1705 | . 2 ⊢ (𝑦 = 𝐴 → (∀𝑥(𝑥 ∈ 𝐵 → 𝑦 ∈ 𝑥) ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝐴 ∈ 𝑥))) |
5 | df-int 3616 | . 2 ⊢ ∩ 𝐵 = {𝑦 ∣ ∀𝑥(𝑥 ∈ 𝐵 → 𝑦 ∈ 𝑥)} | |
6 | 1, 4, 5 | elab2 2690 | 1 ⊢ (𝐴 ∈ ∩ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝐴 ∈ 𝑥)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 ∀wal 1241 = wceq 1243 ∈ wcel 1393 Vcvv 2557 ∩ cint 3615 |
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 |
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-int 3616 |
This theorem is referenced by: elint2 3622 elintab 3626 intss1 3630 intss 3636 intun 3646 intpr 3647 peano1 4317 |
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