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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 29332 | . 2 ⊢ 𝐴 ∈ Sℋ |
3 | chjcl.2 | . . 3 ⊢ 𝐵 ∈ Cℋ | |
4 | 3 | chshii 29332 | . 2 ⊢ 𝐵 ∈ Sℋ |
5 | 2, 4 | shjcomi 29476 | 1 ⊢ (𝐴 ∨ℋ 𝐵) = (𝐵 ∨ℋ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1543 ∈ wcel 2111 (class class class)co 7231 Cℋ cch 29034 ∨ℋ chj 29038 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-sep 5206 ax-nul 5213 ax-pr 5336 ax-hilex 29104 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ral 3067 df-rex 3068 df-rab 3071 df-v 3422 df-sbc 3709 df-dif 3883 df-un 3885 df-in 3887 df-ss 3897 df-nul 4252 df-if 4454 df-pw 4529 df-sn 4556 df-pr 4558 df-op 4562 df-uni 4834 df-br 5068 df-opab 5130 df-id 5469 df-xp 5571 df-rel 5572 df-cnv 5573 df-co 5574 df-dm 5575 df-rn 5576 df-res 5577 df-ima 5578 df-iota 6355 df-fun 6399 df-fv 6405 df-ov 7234 df-oprab 7235 df-mpo 7236 df-sh 29312 df-ch 29326 df-chj 29415 |
This theorem is referenced by: chub2i 29575 chnlei 29590 chj12i 29627 lejdiri 29644 cmcm2i 29698 cmbr3i 29705 qlax2i 29733 osumcor2i 29749 3oalem5 29771 pjcji 29789 mayetes3i 29834 mdslj2i 30425 mdsl1i 30426 cvmdi 30429 mdslmd2i 30435 mdexchi 30440 cvexchi 30474 atabsi 30506 mdsymlem1 30508 mdsymlem6 30513 mdsymlem8 30515 sumdmdlem2 30524 dmdbr5ati 30527 |
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