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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 4844 | . 2 ⊢ (𝐴 ∈ ∩ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝐴 ∈ 𝑥)) |
3 | df-ral 3111 | . 2 ⊢ (∀𝑥 ∈ 𝐵 𝐴 ∈ 𝑥 ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝐴 ∈ 𝑥)) | |
4 | 2, 3 | bitr4i 281 | 1 ⊢ (𝐴 ∈ ∩ 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∀wal 1536 ∈ wcel 2111 ∀wral 3106 Vcvv 3441 ∩ cint 4838 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-ral 3111 df-int 4839 |
This theorem is referenced by: int0 4852 ssint 4854 intssuni 4860 iinuni 4983 onint 7490 intwun 10146 inttsk 10185 intgru 10225 subgint 18295 subrgint 19550 lssintcl 19729 toponmre 21698 alexsubALTlem3 22654 shintcli 29112 chintcli 29114 intlidl 31010 fin2so 35044 intidl 35467 mzpincl 39675 elimaint 40349 elintima 40354 intsal 42970 salgencntex 42983 |
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