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Theorem cotr 5026
 Description: Two ways of saying a relation is transitive. Definition of transitivity in [Schechter] p. 51. (The proof was shortened by Andrew Salmon, 27-Aug-2011.) (Contributed by set.mm contributors, 27-Dec-1996.) (Revised by set.mm contributors, 27-Aug-2011.)
Assertion
Ref Expression
cotr ((R R) Rxyz((xRy yRz) → xRz))
Distinct variable group:   x,y,z,R

Proof of Theorem cotr
StepHypRef Expression
1 ssrel 4844 . 2 ((R R) Rxz(x, z (R R) → x, z R))
2 alcom 1737 . . . 4 (yz((xRy yRz) → xRz) ↔ zy((xRy yRz) → xRz))
3 19.23v 1891 . . . . . 6 (y((xRy yRz) → xRz) ↔ (y(xRy yRz) → xRz))
4 brco 4883 . . . . . . . 8 (x(R R)zy(xRy yRz))
5 df-br 4640 . . . . . . . 8 (x(R R)zx, z (R R))
64, 5bitr3i 242 . . . . . . 7 (y(xRy yRz) ↔ x, z (R R))
7 df-br 4640 . . . . . . 7 (xRzx, z R)
86, 7imbi12i 316 . . . . . 6 ((y(xRy yRz) → xRz) ↔ (x, z (R R) → x, z R))
93, 8bitri 240 . . . . 5 (y((xRy yRz) → xRz) ↔ (x, z (R R) → x, z R))
109albii 1566 . . . 4 (zy((xRy yRz) → xRz) ↔ z(x, z (R R) → x, z R))
112, 10bitri 240 . . 3 (yz((xRy yRz) → xRz) ↔ z(x, z (R R) → x, z R))
1211albii 1566 . 2 (xyz((xRy yRz) → xRz) ↔ xz(x, z (R R) → x, z R))
131, 12bitr4i 243 1 ((R R) Rxyz((xRy yRz) → xRz))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃wex 1541   ∈ wcel 1710   ⊆ wss 3257  ⟨cop 4561   class class class wbr 4639   ∘ ccom 4721 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-co 4726 This theorem is referenced by: (None)
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