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Mirrors > Home > HSE Home > Th. List > chjcomi | Structured version Visualization version GIF version |
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 28656 | . 2 ⊢ 𝐴 ∈ Sℋ |
3 | chjcl.2 | . . 3 ⊢ 𝐵 ∈ Cℋ | |
4 | 3 | chshii 28656 | . 2 ⊢ 𝐵 ∈ Sℋ |
5 | 2, 4 | shjcomi 28802 | 1 ⊢ (𝐴 ∨ℋ 𝐵) = (𝐵 ∨ℋ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1601 ∈ wcel 2106 (class class class)co 6922 Cℋ cch 28358 ∨ℋ chj 28362 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-sep 5017 ax-nul 5025 ax-pr 5138 ax-hilex 28428 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3399 df-sbc 3652 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4672 df-br 4887 df-opab 4949 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-iota 6099 df-fun 6137 df-fv 6143 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-sh 28636 df-ch 28650 df-chj 28741 |
This theorem is referenced by: chub2i 28901 chnlei 28916 chj12i 28953 lejdiri 28970 cmcm2i 29024 cmbr3i 29031 qlax2i 29059 osumcor2i 29075 3oalem5 29097 pjcji 29115 mayetes3i 29160 mdslj2i 29751 mdsl1i 29752 cvmdi 29755 mdslmd2i 29761 mdexchi 29766 cvexchi 29800 atabsi 29832 mdsymlem1 29834 mdsymlem6 29839 mdsymlem8 29841 sumdmdlem2 29850 dmdbr5ati 29853 |
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