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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 4882 | . 2 ⊢ (𝐴 ∈ ∩ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝐴 ∈ 𝑥)) |
3 | df-ral 3143 | . 2 ⊢ (∀𝑥 ∈ 𝐵 𝐴 ∈ 𝑥 ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝐴 ∈ 𝑥)) | |
4 | 2, 3 | bitr4i 280 | 1 ⊢ (𝐴 ∈ ∩ 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∀wal 1535 ∈ wcel 2114 ∀wral 3138 Vcvv 3494 ∩ cint 4876 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-int 4877 |
This theorem is referenced by: int0 4890 ssint 4892 intssuni 4898 iinuni 5020 onint 7510 intwun 10157 inttsk 10196 intgru 10236 subgint 18303 subrgint 19557 lssintcl 19736 toponmre 21701 alexsubALTlem3 22657 shintcli 29106 chintcli 29108 fin2so 34894 intidl 35322 mzpincl 39351 elimaint 40013 elintima 40018 intsal 42633 salgencntex 42646 |
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