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Mirrors > Home > HSE Home > Th. List > shincli | Structured version Visualization version GIF version |
Description: Closure of intersection of two subspaces. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.) |
Ref | Expression |
---|---|
shincl.1 | ⊢ 𝐴 ∈ Sℋ |
shincl.2 | ⊢ 𝐵 ∈ Sℋ |
Ref | Expression |
---|---|
shincli | ⊢ (𝐴 ∩ 𝐵) ∈ Sℋ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | shincl.1 | . . . 4 ⊢ 𝐴 ∈ Sℋ | |
2 | 1 | elexi 3186 | . . 3 ⊢ 𝐴 ∈ V |
3 | shincl.2 | . . . 4 ⊢ 𝐵 ∈ Sℋ | |
4 | 3 | elexi 3186 | . . 3 ⊢ 𝐵 ∈ V |
5 | 2, 4 | intpr 4445 | . 2 ⊢ ∩ {𝐴, 𝐵} = (𝐴 ∩ 𝐵) |
6 | 1, 3 | pm3.2i 470 | . . . . 5 ⊢ (𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) |
7 | 2, 4 | prss 4291 | . . . . 5 ⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) ↔ {𝐴, 𝐵} ⊆ Sℋ ) |
8 | 6, 7 | mpbi 219 | . . . 4 ⊢ {𝐴, 𝐵} ⊆ Sℋ |
9 | 2 | prnz 4253 | . . . 4 ⊢ {𝐴, 𝐵} ≠ ∅ |
10 | 8, 9 | pm3.2i 470 | . . 3 ⊢ ({𝐴, 𝐵} ⊆ Sℋ ∧ {𝐴, 𝐵} ≠ ∅) |
11 | 10 | shintcli 27572 | . 2 ⊢ ∩ {𝐴, 𝐵} ∈ Sℋ |
12 | 5, 11 | eqeltrri 2685 | 1 ⊢ (𝐴 ∩ 𝐵) ∈ Sℋ |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 ∈ wcel 1977 ≠ wne 2780 ∩ cin 3539 ⊆ wss 3540 ∅c0 3874 {cpr 4127 ∩ cint 4410 Sℋ csh 27169 |
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-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 ax-hilex 27240 ax-hfvadd 27241 ax-hv0cl 27244 ax-hfvmul 27246 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 df-ov 6552 df-sh 27448 |
This theorem is referenced by: shincl 27624 shmodsi 27632 shmodi 27633 5oalem1 27897 5oalem3 27899 5oalem5 27901 5oalem6 27902 5oai 27904 3oalem2 27906 3oalem6 27910 cdj3lem1 28677 |
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