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Theorem ertr 6057
Description: An equivalence relation is transitive. (Contributed by NM, 4-Jun-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)
Hypothesis
Ref Expression
ersymb.1 (φ𝑅 Er 𝑋)
Assertion
Ref Expression
ertr (φ → ((A𝑅B B𝑅𝐶) → A𝑅𝐶))

Proof of Theorem ertr
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 ersymb.1 . . . . . . 7 (φ𝑅 Er 𝑋)
2 errel 6051 . . . . . . 7 (𝑅 Er 𝑋 → Rel 𝑅)
31, 2syl 14 . . . . . 6 (φ → Rel 𝑅)
4 simpr 103 . . . . . 6 ((A𝑅B B𝑅𝐶) → B𝑅𝐶)
5 brrelex 4325 . . . . . 6 ((Rel 𝑅 B𝑅𝐶) → B V)
63, 4, 5syl2an 273 . . . . 5 ((φ (A𝑅B B𝑅𝐶)) → B V)
7 simpr 103 . . . . 5 ((φ (A𝑅B B𝑅𝐶)) → (A𝑅B B𝑅𝐶))
8 breq2 3759 . . . . . . 7 (x = B → (A𝑅xA𝑅B))
9 breq1 3758 . . . . . . 7 (x = B → (x𝑅𝐶B𝑅𝐶))
108, 9anbi12d 442 . . . . . 6 (x = B → ((A𝑅x x𝑅𝐶) ↔ (A𝑅B B𝑅𝐶)))
1110spcegv 2635 . . . . 5 (B V → ((A𝑅B B𝑅𝐶) → x(A𝑅x x𝑅𝐶)))
126, 7, 11sylc 56 . . . 4 ((φ (A𝑅B B𝑅𝐶)) → x(A𝑅x x𝑅𝐶))
13 simpl 102 . . . . . 6 ((A𝑅B B𝑅𝐶) → A𝑅B)
14 brrelex 4325 . . . . . 6 ((Rel 𝑅 A𝑅B) → A V)
153, 13, 14syl2an 273 . . . . 5 ((φ (A𝑅B B𝑅𝐶)) → A V)
16 brrelex2 4326 . . . . . 6 ((Rel 𝑅 B𝑅𝐶) → 𝐶 V)
173, 4, 16syl2an 273 . . . . 5 ((φ (A𝑅B B𝑅𝐶)) → 𝐶 V)
18 brcog 4445 . . . . 5 ((A V 𝐶 V) → (A(𝑅𝑅)𝐶x(A𝑅x x𝑅𝐶)))
1915, 17, 18syl2anc 391 . . . 4 ((φ (A𝑅B B𝑅𝐶)) → (A(𝑅𝑅)𝐶x(A𝑅x x𝑅𝐶)))
2012, 19mpbird 156 . . 3 ((φ (A𝑅B B𝑅𝐶)) → A(𝑅𝑅)𝐶)
2120ex 108 . 2 (φ → ((A𝑅B B𝑅𝐶) → A(𝑅𝑅)𝐶))
22 df-er 6042 . . . . . 6 (𝑅 Er 𝑋 ↔ (Rel 𝑅 dom 𝑅 = 𝑋 (𝑅 ∪ (𝑅𝑅)) ⊆ 𝑅))
2322simp3bi 920 . . . . 5 (𝑅 Er 𝑋 → (𝑅 ∪ (𝑅𝑅)) ⊆ 𝑅)
241, 23syl 14 . . . 4 (φ → (𝑅 ∪ (𝑅𝑅)) ⊆ 𝑅)
2524unssbd 3115 . . 3 (φ → (𝑅𝑅) ⊆ 𝑅)
2625ssbrd 3796 . 2 (φ → (A(𝑅𝑅)𝐶A𝑅𝐶))
2721, 26syld 40 1 (φ → ((A𝑅B B𝑅𝐶) → A𝑅𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242  wex 1378   wcel 1390  Vcvv 2551  cun 2909  wss 2911   class class class wbr 3755  ccnv 4287  dom cdm 4288  ccom 4292  Rel wrel 4293   Er wer 6039
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-co 4297  df-er 6042
This theorem is referenced by:  ertrd  6058  erth  6086  iinerm  6114  entr  6200
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