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Theorem shjcomi 31201
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 31188 . 2 ((𝐴S𝐵S ) → (𝐴 𝐵) = (𝐵 𝐴))
41, 2, 3mp2an 690 1 (𝐴 𝐵) = (𝐵 𝐴)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533  wcel 2098  (class class class)co 7426   S csh 30758   chj 30763
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-hilex 30829
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-sh 31037  df-chj 31140
This theorem is referenced by:  shlej2i  31209  chjcomi  31298
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