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Mirrors > Home > HSE Home > Th. List > shjcomi | Structured version Visualization version GIF version |
Description: Commutative law for join in Sℋ. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.) |
Ref | Expression |
---|---|
shincl.1 | ⊢ 𝐴 ∈ Sℋ |
shincl.2 | ⊢ 𝐵 ∈ Sℋ |
Ref | Expression |
---|---|
shjcomi | ⊢ (𝐴 ∨ℋ 𝐵) = (𝐵 ∨ℋ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | shincl.1 | . 2 ⊢ 𝐴 ∈ Sℋ | |
2 | shincl.2 | . 2 ⊢ 𝐵 ∈ Sℋ | |
3 | shjcom 28217 | . 2 ⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → (𝐴 ∨ℋ 𝐵) = (𝐵 ∨ℋ 𝐴)) | |
4 | 1, 2, 3 | mp2an 708 | 1 ⊢ (𝐴 ∨ℋ 𝐵) = (𝐵 ∨ℋ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1483 ∈ wcel 1990 (class class class)co 6650 Sℋ csh 27785 ∨ℋ chj 27790 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 ax-hilex 27856 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-sh 28064 df-chj 28169 |
This theorem is referenced by: shlej2i 28238 chjcomi 28327 |
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