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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 3511 | . . 3 ⊢ 𝐴 ∈ V |
3 | shincl.2 | . . . 4 ⊢ 𝐵 ∈ Sℋ | |
4 | 3 | elexi 3511 | . . 3 ⊢ 𝐵 ∈ V |
5 | 2, 4 | intpr 5006 | . 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 31361 | . 2 ⊢ ∩ {𝐴, 𝐵} ∈ Sℋ |
12 | 5, 11 | eqeltrri 2841 | 1 ⊢ (𝐴 ∩ 𝐵) ∈ Sℋ |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 ∈ wcel 2108 ≠ wne 2946 ∩ cin 3975 ⊆ wss 3976 ∅c0 4352 {cpr 4650 ∩ cint 4970 Sℋ csh 30960 |
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 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pr 5447 ax-hilex 31031 ax-hfvadd 31032 ax-hv0cl 31035 ax-hfvmul 31037 |
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 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-ral 3068 df-rex 3077 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-br 5167 df-opab 5229 df-id 5593 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-fv 6581 df-ov 7451 df-sh 31239 |
This theorem is referenced by: shincl 31413 shmodsi 31421 shmodi 31422 5oalem1 31686 5oalem3 31688 5oalem5 31690 5oalem6 31691 5oai 31693 3oalem2 31695 3oalem6 31699 cdj3lem1 32466 |
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