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Theorem coemptyd 14200
Description: Deduction about composition of classes with no relational content in common. (Contributed by RP, 24-Dec-2019.)
Hypothesis
Ref Expression
coemptyd.1 (𝜑 → (dom 𝐴 ∩ ran 𝐵) = ∅)
Assertion
Ref Expression
coemptyd (𝜑 → (𝐴𝐵) = ∅)

Proof of Theorem coemptyd
StepHypRef Expression
1 coemptyd.1 . 2 (𝜑 → (dom 𝐴 ∩ ran 𝐵) = ∅)
2 coeq0 5947 . 2 ((𝐴𝐵) = ∅ ↔ (dom 𝐴 ∩ ran 𝐵) = ∅)
31, 2sylibr 226 1 (𝜑 → (𝐴𝐵) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1507  cin 3828  c0 4178  dom cdm 5407  ran crn 5408  ccom 5411
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-sep 5060  ax-nul 5067  ax-pr 5186
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ral 3093  df-rex 3094  df-rab 3097  df-v 3417  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-nul 4179  df-if 4351  df-sn 4442  df-pr 4444  df-op 4448  df-br 4930  df-opab 4992  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419
This theorem is referenced by:  xptrrel  14201  cosnopne  30189  coeq0i  38751  conrel1d  39377  conrel2d  39378  clsneibex  39821  neicvgbex  39831
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