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Theorem shjval 29137
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 28996 . 2 (𝐴S𝐴 ⊆ ℋ)
2 shss 28996 . 2 (𝐵S𝐵 ⊆ ℋ)
3 sshjval 29136 . 2 ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (𝐴 𝐵) = (⊥‘(⊥‘(𝐴𝐵))))
41, 2, 3syl2an 598 1 ((𝐴S𝐵S ) → (𝐴 𝐵) = (⊥‘(⊥‘(𝐴𝐵))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2112  cun 3882  wss 3884  cfv 6328  (class class class)co 7139  chba 28705   S csh 28714  cort 28716   chj 28719
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298  ax-hilex 28785
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-sh 28993  df-chj 29096
This theorem is referenced by:  chjval  29138  shjcom  29144  shlej1  29146  shunssji  29155  shlub  29200  shjshsi  29278
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