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Mirrors > Home > HSE Home > Th. List > chm1i | Structured version Visualization version GIF version |
Description: Meet with lattice one in Cℋ. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ch0le.1 | ⊢ 𝐴 ∈ Cℋ |
Ref | Expression |
---|---|
chm1i | ⊢ (𝐴 ∩ ℋ) = 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ch0le.1 | . . 3 ⊢ 𝐴 ∈ Cℋ | |
2 | 1 | chssii 28787 | . 2 ⊢ 𝐴 ⊆ ℋ |
3 | df-ss 3844 | . 2 ⊢ (𝐴 ⊆ ℋ ↔ (𝐴 ∩ ℋ) = 𝐴) | |
4 | 2, 3 | mpbi 222 | 1 ⊢ (𝐴 ∩ ℋ) = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1507 ∈ wcel 2050 ∩ cin 3829 ⊆ wss 3830 ℋchba 28475 Cℋ cch 28485 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-ext 2751 ax-sep 5060 ax-hilex 28555 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-clab 2760 df-cleq 2772 df-clel 2847 df-nfc 2919 df-rex 3095 df-rab 3098 df-v 3418 df-dif 3833 df-un 3835 df-in 3837 df-ss 3844 df-nul 4180 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-op 4448 df-uni 4713 df-br 4930 df-opab 4992 df-xp 5413 df-cnv 5415 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-iota 6152 df-fv 6196 df-ov 6979 df-sh 28763 df-ch 28777 |
This theorem is referenced by: stcltrlem1 29834 |
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