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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 3506 | . . 3 ⊢ 𝐴 ∈ V |
3 | shincl.2 | . . . 4 ⊢ 𝐵 ∈ Sℋ | |
4 | 3 | elexi 3506 | . . 3 ⊢ 𝐵 ∈ V |
5 | 2, 4 | intpr 5008 | . 2 ⊢ ∩ {𝐴, 𝐵} = (𝐴 ∩ 𝐵) |
6 | 1, 3 | pm3.2i 470 | . . . . 5 ⊢ (𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) |
7 | 2, 4 | prss 4845 | . . . . 5 ⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) ↔ {𝐴, 𝐵} ⊆ Sℋ ) |
8 | 6, 7 | mpbi 230 | . . . 4 ⊢ {𝐴, 𝐵} ⊆ Sℋ |
9 | 2 | prnz 4802 | . . . 4 ⊢ {𝐴, 𝐵} ≠ ∅ |
10 | 8, 9 | pm3.2i 470 | . . 3 ⊢ ({𝐴, 𝐵} ⊆ Sℋ ∧ {𝐴, 𝐵} ≠ ∅) |
11 | 10 | shintcli 31352 | . 2 ⊢ ∩ {𝐴, 𝐵} ∈ Sℋ |
12 | 5, 11 | eqeltrri 2835 | 1 ⊢ (𝐴 ∩ 𝐵) ∈ Sℋ |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 ∈ wcel 2103 ≠ wne 2942 ∩ cin 3969 ⊆ wss 3970 ∅c0 4347 {cpr 4650 ∩ cint 4972 Sℋ csh 30951 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-sep 5320 ax-nul 5327 ax-pr 5450 ax-hilex 31022 ax-hfvadd 31023 ax-hv0cl 31026 ax-hfvmul 31028 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ne 2943 df-ral 3064 df-rex 3073 df-rab 3439 df-v 3484 df-sbc 3799 df-csb 3916 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-nul 4348 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-int 4973 df-iun 5021 df-br 5170 df-opab 5232 df-id 5597 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-res 5711 df-ima 5712 df-iota 6524 df-fun 6574 df-fn 6575 df-f 6576 df-fv 6580 df-ov 7448 df-sh 31230 |
This theorem is referenced by: shincl 31404 shmodsi 31412 shmodi 31413 5oalem1 31677 5oalem3 31679 5oalem5 31681 5oalem6 31682 5oai 31684 3oalem2 31686 3oalem6 31690 cdj3lem1 32457 |
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