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Theorem shjval 28766
Description: Value of join in S. (Contributed by NM, 9-Aug-2000.) (New usage is discouraged.)
Assertion
Ref Expression
shjval ((𝐴S𝐵S ) → (𝐴 𝐵) = (⊥‘(⊥‘(𝐴𝐵))))

Proof of Theorem shjval
StepHypRef Expression
1 shss 28623 . 2 (𝐴S𝐴 ⊆ ℋ)
2 shss 28623 . 2 (𝐵S𝐵 ⊆ ℋ)
3 sshjval 28765 . 2 ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (𝐴 𝐵) = (⊥‘(⊥‘(𝐴𝐵))))
41, 2, 3syl2an 591 1 ((𝐴S𝐵S ) → (𝐴 𝐵) = (⊥‘(⊥‘(𝐴𝐵))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1658  wcel 2166  cun 3797  wss 3799  cfv 6124  (class class class)co 6906  chba 28332   S csh 28341  cort 28343   chj 28346
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-sep 5006  ax-nul 5014  ax-pr 5128  ax-hilex 28412
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ral 3123  df-rex 3124  df-rab 3127  df-v 3417  df-sbc 3664  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4660  df-br 4875  df-opab 4937  df-id 5251  df-xp 5349  df-rel 5350  df-cnv 5351  df-co 5352  df-dm 5353  df-rn 5354  df-res 5355  df-ima 5356  df-iota 6087  df-fun 6126  df-fv 6132  df-ov 6909  df-oprab 6910  df-mpt2 6911  df-sh 28620  df-chj 28725
This theorem is referenced by:  chjval  28767  shjcom  28773  shlej1  28775  shunssji  28784  shlub  28829  shjshsi  28907
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