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Mirrors > Home > ILE Home > Th. List > elint2 | GIF version |
Description: Membership in class intersection. (Contributed by NM, 14-Oct-1999.) |
Ref | Expression |
---|---|
elint2.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
elint2 | ⊢ (𝐴 ∈ ∩ 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elint2.1 | . . 3 ⊢ 𝐴 ∈ V | |
2 | 1 | elint 3777 | . 2 ⊢ (𝐴 ∈ ∩ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝐴 ∈ 𝑥)) |
3 | df-ral 2421 | . 2 ⊢ (∀𝑥 ∈ 𝐵 𝐴 ∈ 𝑥 ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝐴 ∈ 𝑥)) | |
4 | 2, 3 | bitr4i 186 | 1 ⊢ (𝐴 ∈ ∩ 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝑥) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∀wal 1329 ∈ wcel 1480 ∀wral 2416 Vcvv 2686 ∩ cint 3771 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-v 2688 df-int 3772 |
This theorem is referenced by: elintg 3779 ssint 3787 intssunim 3793 iinuniss 3895 trint 4041 suplocexprlemmu 7526 suplocexprlemdisj 7528 suplocexprlemloc 7529 suplocexprlemub 7531 |
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