Theorem shjcomi 29150

Description: Commutative law for join in S_ℋ. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)

Hypotheses

Ref	Expression
shincl.1	⊢ 𝐴 ∈ Sℋ
shincl.2	⊢ 𝐵 ∈ Sℋ

Assertion

Ref	Expression
shjcomi	⊢ (𝐴 ∨ℋ 𝐵) = (𝐵 ∨ℋ 𝐴)

Proof of Theorem shjcomi

Step	Hyp	Ref	Expression
1		shincl.1	. 2 ⊢ 𝐴 ∈ Sℋ
2		shincl.2	. 2 ⊢ 𝐵 ∈ Sℋ
3		shjcom 29137	. 2 ⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → (𝐴 ∨ℋ 𝐵) = (𝐵 ∨ℋ 𝐴))
4	1, 2, 3	mp2an 690	1 ⊢ (𝐴 ∨ℋ 𝐵) = (𝐵 ∨ℋ 𝐴)

Syntax hints:   = wceq 1537   ∈ wcel 2114  (class class class)co 7158   S_ℋ csh 28707   ∨_ℋ chj 28712

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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332  ax-hilex 28778

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-iota 6316  df-fun 6359  df-fv 6365  df-ov 7161  df-oprab 7162  df-mpo 7163  df-sh 28986  df-chj 29089

This theorem is referenced by:  shlej2i  29158  chjcomi  29247