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| Description: Value of join in Cℋ. (Contributed by NM, 9-Aug-2000.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| chjval.1 | ⊢ 𝐴 ∈ Cℋ | 
| chjval.2 | ⊢ 𝐵 ∈ Cℋ | 
| Ref | Expression | 
|---|---|
| chjvali | ⊢ (𝐴 ∨ℋ 𝐵) = (⊥‘(⊥‘(𝐴 ∪ 𝐵))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | chjval.1 | . 2 ⊢ 𝐴 ∈ Cℋ | |
| 2 | chjval.2 | . 2 ⊢ 𝐵 ∈ Cℋ | |
| 3 | chjval 31372 | . 2 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ∨ℋ 𝐵) = (⊥‘(⊥‘(𝐴 ∪ 𝐵)))) | |
| 4 | 1, 2, 3 | mp2an 692 | 1 ⊢ (𝐴 ∨ℋ 𝐵) = (⊥‘(⊥‘(𝐴 ∪ 𝐵))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1539 ∈ wcel 2107 ∪ cun 3948 ‘cfv 6560 (class class class)co 7432 Cℋ cch 30949 ⊥cort 30950 ∨ℋ chj 30953 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 ax-hilex 31019 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-sh 31227 df-ch 31241 df-chj 31330 | 
| This theorem is referenced by: chj0i 31475 sshhococi 31566 | 
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