New Foundations Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > NFE 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 2413 | . . . 4 ⊢ (y = A → (y ∈ x ↔ A ∈ x)) | |
3 | 2 | imbi2d 307 | . . 3 ⊢ (y = A → ((x ∈ B → y ∈ x) ↔ (x ∈ B → A ∈ x))) |
4 | 3 | albidv 1625 | . 2 ⊢ (y = A → (∀x(x ∈ B → y ∈ x) ↔ ∀x(x ∈ B → A ∈ x))) |
5 | df-int 3927 | . 2 ⊢ ∩B = {y ∣ ∀x(x ∈ B → y ∈ x)} | |
6 | 1, 4, 5 | elab2 2988 | 1 ⊢ (A ∈ ∩B ↔ ∀x(x ∈ B → A ∈ x)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 ∀wal 1540 = wceq 1642 ∈ wcel 1710 Vcvv 2859 ∩cint 3926 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-v 2861 df-int 3927 |
This theorem is referenced by: elint2 3933 elintab 3937 intss1 3941 intss 3947 intun 3958 intpr 3959 |
Copyright terms: Public domain | W3C validator |