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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 3213 | . . 3 ⊢ 𝐴 ∈ V |
3 | shincl.2 | . . . 4 ⊢ 𝐵 ∈ Sℋ | |
4 | 3 | elexi 3213 | . . 3 ⊢ 𝐵 ∈ V |
5 | 2, 4 | intpr 4510 | . 2 ⊢ ∩ {𝐴, 𝐵} = (𝐴 ∩ 𝐵) |
6 | 1, 3 | pm3.2i 471 | . . . . 5 ⊢ (𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) |
7 | 2, 4 | prss 4351 | . . . . 5 ⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) ↔ {𝐴, 𝐵} ⊆ Sℋ ) |
8 | 6, 7 | mpbi 220 | . . . 4 ⊢ {𝐴, 𝐵} ⊆ Sℋ |
9 | 2 | prnz 4310 | . . . 4 ⊢ {𝐴, 𝐵} ≠ ∅ |
10 | 8, 9 | pm3.2i 471 | . . 3 ⊢ ({𝐴, 𝐵} ⊆ Sℋ ∧ {𝐴, 𝐵} ≠ ∅) |
11 | 10 | shintcli 28188 | . 2 ⊢ ∩ {𝐴, 𝐵} ∈ Sℋ |
12 | 5, 11 | eqeltrri 2698 | 1 ⊢ (𝐴 ∩ 𝐵) ∈ Sℋ |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 384 ∈ wcel 1990 ≠ wne 2794 ∩ cin 3573 ⊆ wss 3574 ∅c0 3915 {cpr 4179 ∩ cint 4475 Sℋ csh 27785 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 ax-hilex 27856 ax-hfvadd 27857 ax-hv0cl 27860 ax-hfvmul 27862 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fv 5896 df-ov 6653 df-sh 28064 |
This theorem is referenced by: shincl 28240 shmodsi 28248 shmodi 28249 5oalem1 28513 5oalem3 28515 5oalem5 28517 5oalem6 28518 5oai 28520 3oalem2 28522 3oalem6 28526 cdj3lem1 29293 |
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