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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 30912 | . 2 ⊢ 𝐴 ∈ Sℋ |
3 | chjcl.2 | . . 3 ⊢ 𝐵 ∈ Cℋ | |
4 | 3 | chshii 30912 | . 2 ⊢ 𝐵 ∈ Sℋ |
5 | 2, 4 | shjcomi 31056 | 1 ⊢ (𝐴 ∨ℋ 𝐵) = (𝐵 ∨ℋ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1540 ∈ wcel 2105 (class class class)co 7412 Cℋ cch 30614 ∨ℋ chj 30618 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427 ax-hilex 30684 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-sbc 3778 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fv 6551 df-ov 7415 df-oprab 7416 df-mpo 7417 df-sh 30892 df-ch 30906 df-chj 30995 |
This theorem is referenced by: chub2i 31155 chnlei 31170 chj12i 31207 lejdiri 31224 cmcm2i 31278 cmbr3i 31285 qlax2i 31313 osumcor2i 31329 3oalem5 31351 pjcji 31369 mayetes3i 31414 mdslj2i 32005 mdsl1i 32006 cvmdi 32009 mdslmd2i 32015 mdexchi 32020 cvexchi 32054 atabsi 32086 mdsymlem1 32088 mdsymlem6 32093 mdsymlem8 32095 sumdmdlem2 32104 dmdbr5ati 32107 |
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