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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 3642 | . 2 ⊢ (𝐴 ∈ ∩ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝐴 ∈ 𝑥)) |
3 | df-ral 2353 | . 2 ⊢ (∀𝑥 ∈ 𝐵 𝐴 ∈ 𝑥 ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝐴 ∈ 𝑥)) | |
4 | 2, 3 | bitr4i 185 | 1 ⊢ (𝐴 ∈ ∩ 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝑥) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 103 ∀wal 1282 ∈ wcel 1433 ∀wral 2348 Vcvv 2601 ∩ cint 3636 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-v 2603 df-int 3637 |
This theorem is referenced by: elintg 3644 ssint 3652 intssunim 3658 iinuniss 3758 trint 3890 trintssmOLD 3892 |
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