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| Description: Commutative law for join in Cℋ. (Contributed by NM, 14-Oct-1999.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| ch0le.1 | ⊢ 𝐴 ∈ Cℋ | 
| chjcl.2 | ⊢ 𝐵 ∈ Cℋ | 
| Ref | Expression | 
|---|---|
| chjcomi | ⊢ (𝐴 ∨ℋ 𝐵) = (𝐵 ∨ℋ 𝐴) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ch0le.1 | . . 3 ⊢ 𝐴 ∈ Cℋ | |
| 2 | 1 | chshii 31246 | . 2 ⊢ 𝐴 ∈ Sℋ | 
| 3 | chjcl.2 | . . 3 ⊢ 𝐵 ∈ Cℋ | |
| 4 | 3 | chshii 31246 | . 2 ⊢ 𝐵 ∈ Sℋ | 
| 5 | 2, 4 | shjcomi 31390 | 1 ⊢ (𝐴 ∨ℋ 𝐵) = (𝐵 ∨ℋ 𝐴) | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1540 ∈ wcel 2108 (class class class)co 7431 Cℋ cch 30948 ∨ℋ chj 30952 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-hilex 31018 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-sh 31226 df-ch 31240 df-chj 31329 | 
| This theorem is referenced by: chub2i 31489 chnlei 31504 chj12i 31541 lejdiri 31558 cmcm2i 31612 cmbr3i 31619 qlax2i 31647 osumcor2i 31663 3oalem5 31685 pjcji 31703 mayetes3i 31748 mdslj2i 32339 mdsl1i 32340 cvmdi 32343 mdslmd2i 32349 mdexchi 32354 cvexchi 32388 atabsi 32420 mdsymlem1 32422 mdsymlem6 32427 mdsymlem8 32429 sumdmdlem2 32438 dmdbr5ati 32441 | 
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