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Theorem shjcomi 28802
Description: Commutative law for join in S. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
Hypotheses
Ref Expression
shincl.1 𝐴S
shincl.2 𝐵S
Assertion
Ref Expression
shjcomi (𝐴 𝐵) = (𝐵 𝐴)

Proof of Theorem shjcomi
StepHypRef Expression
1 shincl.1 . 2 𝐴S
2 shincl.2 . 2 𝐵S
3 shjcom 28789 . 2 ((𝐴S𝐵S ) → (𝐴 𝐵) = (𝐵 𝐴))
41, 2, 3mp2an 682 1 (𝐴 𝐵) = (𝐵 𝐴)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1601  wcel 2106  (class class class)co 6922   S csh 28357   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-chj 28741
This theorem is referenced by:  shlej2i  28810  chjcomi  28899
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