Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 2180 | . . . 4 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ 𝑥 ↔ 𝐴 ∈ 𝑥)) | |
3 | 2 | imbi2d 229 | . . 3 ⊢ (𝑦 = 𝐴 → ((𝑥 ∈ 𝐵 → 𝑦 ∈ 𝑥) ↔ (𝑥 ∈ 𝐵 → 𝐴 ∈ 𝑥))) |
4 | 3 | albidv 1780 | . 2 ⊢ (𝑦 = 𝐴 → (∀𝑥(𝑥 ∈ 𝐵 → 𝑦 ∈ 𝑥) ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝐴 ∈ 𝑥))) |
5 | df-int 3742 | . 2 ⊢ ∩ 𝐵 = {𝑦 ∣ ∀𝑥(𝑥 ∈ 𝐵 → 𝑦 ∈ 𝑥)} | |
6 | 1, 4, 5 | elab2 2805 | 1 ⊢ (𝐴 ∈ ∩ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝐴 ∈ 𝑥)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∀wal 1314 = wceq 1316 ∈ wcel 1465 Vcvv 2660 ∩ cint 3741 |
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-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-v 2662 df-int 3742 |
This theorem is referenced by: elint2 3748 elintab 3752 intss1 3756 intss 3762 intun 3772 intpr 3773 peano1 4478 |
Copyright terms: Public domain | W3C validator |