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Theorem trrelind 36328
 Description: The intersection of transitive relations is a transitive relation. (Contributed by Richard Penner, 24-Dec-2019.)
Hypotheses
Ref Expression
trrelind.r
trrelind.s
trrelind.t
Assertion
Ref Expression
trrelind

Proof of Theorem trrelind
StepHypRef Expression
1 trrelind.r . . . 4
2 inss1 3643 . . . . 5
32a1i 11 . . . 4
41, 3, 3trrelssd 13112 . . 3
5 trrelind.s . . . 4
6 inss2 3644 . . . . 5
76a1i 11 . . . 4
85, 7, 7trrelssd 13112 . . 3
94, 8ssind 3647 . 2
10 trrelind.t . . 3
1110, 10coeq12d 5004 . 2
129, 11, 103sstr4d 3461 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wceq 1452   cin 3389   wss 3390   ccom 4843 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451 This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-v 3033  df-in 3397  df-ss 3404  df-br 4396  df-opab 4455  df-co 4848 This theorem is referenced by:  xpintrreld  36329
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