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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 31034 | . 2 ⊢ 𝐴 ⊆ ℋ |
3 | df-ss 3961 | . 2 ⊢ (𝐴 ⊆ ℋ ↔ (𝐴 ∩ ℋ) = 𝐴) | |
4 | 2, 3 | mpbi 229 | 1 ⊢ (𝐴 ∩ ℋ) = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 ∈ wcel 2099 ∩ cin 3943 ⊆ wss 3944 ℋchba 30722 Cℋ cch 30732 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2698 ax-sep 5293 ax-hilex 30802 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2705 df-cleq 2719 df-clel 2805 df-rab 3428 df-v 3471 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-xp 5678 df-cnv 5680 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fv 6550 df-ov 7417 df-sh 31010 df-ch 31024 |
This theorem is referenced by: stcltrlem1 32079 |
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