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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 3417 | . . 3 ⊢ 𝐴 ∈ V |
3 | shincl.2 | . . . 4 ⊢ 𝐵 ∈ Sℋ | |
4 | 3 | elexi 3417 | . . 3 ⊢ 𝐵 ∈ V |
5 | 2, 4 | intpr 4870 | . 2 ⊢ ∩ {𝐴, 𝐵} = (𝐴 ∩ 𝐵) |
6 | 1, 3 | pm3.2i 474 | . . . . 5 ⊢ (𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) |
7 | 2, 4 | prss 4708 | . . . . 5 ⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) ↔ {𝐴, 𝐵} ⊆ Sℋ ) |
8 | 6, 7 | mpbi 233 | . . . 4 ⊢ {𝐴, 𝐵} ⊆ Sℋ |
9 | 2 | prnz 4668 | . . . 4 ⊢ {𝐴, 𝐵} ≠ ∅ |
10 | 8, 9 | pm3.2i 474 | . . 3 ⊢ ({𝐴, 𝐵} ⊆ Sℋ ∧ {𝐴, 𝐵} ≠ ∅) |
11 | 10 | shintcli 29264 | . 2 ⊢ ∩ {𝐴, 𝐵} ∈ Sℋ |
12 | 5, 11 | eqeltrri 2830 | 1 ⊢ (𝐴 ∩ 𝐵) ∈ Sℋ |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 ∈ wcel 2114 ≠ wne 2934 ∩ cin 3842 ⊆ wss 3843 ∅c0 4211 {cpr 4518 ∩ cint 4836 Sℋ csh 28863 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pr 5296 ax-hilex 28934 ax-hfvadd 28935 ax-hv0cl 28938 ax-hfvmul 28940 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-op 4523 df-uni 4797 df-int 4837 df-iun 4883 df-br 5031 df-opab 5093 df-id 5429 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-fv 6347 df-ov 7173 df-sh 29142 |
This theorem is referenced by: shincl 29316 shmodsi 29324 shmodi 29325 5oalem1 29589 5oalem3 29591 5oalem5 29593 5oalem6 29594 5oai 29596 3oalem2 29598 3oalem6 29602 cdj3lem1 30369 |
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