![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > elint | Structured version Visualization version 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 2822 | . . . . 5 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ 𝑥 ↔ 𝐴 ∈ 𝑥)) | |
3 | 2 | imbi2d 341 | . . . 4 ⊢ (𝑦 = 𝐴 → ((𝑥 ∈ 𝐵 → 𝑦 ∈ 𝑥) ↔ (𝑥 ∈ 𝐵 → 𝐴 ∈ 𝑥))) |
4 | 3 | albidv 1924 | . . 3 ⊢ (𝑦 = 𝐴 → (∀𝑥(𝑥 ∈ 𝐵 → 𝑦 ∈ 𝑥) ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝐴 ∈ 𝑥))) |
5 | df-int 4950 | . . 3 ⊢ ∩ 𝐵 = {𝑦 ∣ ∀𝑥(𝑥 ∈ 𝐵 → 𝑦 ∈ 𝑥)} | |
6 | 4, 5 | elab2g 3669 | . 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ ∩ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝐴 ∈ 𝑥))) |
7 | 1, 6 | ax-mp 5 | 1 ⊢ (𝐴 ∈ ∩ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝐴 ∈ 𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∀wal 1540 = wceq 1542 ∈ wcel 2107 Vcvv 3475 ∩ cint 4949 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-int 4950 |
This theorem is referenced by: elint2 4956 elintabOLD 4962 intss1 4966 intun 4983 intprg 4984 intprOLD 4986 cssmre 21230 elintfv 34674 dfom5b 34822 |
Copyright terms: Public domain | W3C validator |