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Mirrors > Home > MPE Home > Th. List > elint2 | Structured version Visualization version 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 4416 | . 2 ⊢ (𝐴 ∈ ∩ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝐴 ∈ 𝑥)) |
3 | df-ral 2901 | . 2 ⊢ (∀𝑥 ∈ 𝐵 𝐴 ∈ 𝑥 ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝐴 ∈ 𝑥)) | |
4 | 2, 3 | bitr4i 266 | 1 ⊢ (𝐴 ∈ ∩ 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∀wal 1473 ∈ wcel 1977 ∀wral 2896 Vcvv 3173 ∩ cint 4410 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-v 3175 df-int 4411 |
This theorem is referenced by: elintgOLD 4419 int0 4425 ssint 4428 intssuni 4434 iinuni 4545 trint 4696 trintss 4697 onint 6887 intwun 9436 inttsk 9475 intgru 9515 subgint 17441 subrgint 18625 lssintcl 18785 toponmre 20707 alexsubALTlem3 21663 shintcli 27572 chintcli 27574 fin2so 32566 intidl 32998 mzpincl 36315 elimaint 36959 elintima 36964 intsal 39224 salgencntex 39237 |
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