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Theorem ertr 6149
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 → ((𝐴𝑅𝐵𝐵𝑅𝐶) → 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 6143 . . . . . . 7 (𝑅 Er 𝑋 → Rel 𝑅)
31, 2syl 14 . . . . . 6 (𝜑 → Rel 𝑅)
4 simpr 107 . . . . . 6 ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐵𝑅𝐶)
5 brrelex 4407 . . . . . 6 ((Rel 𝑅𝐵𝑅𝐶) → 𝐵 ∈ V)
63, 4, 5syl2an 277 . . . . 5 ((𝜑 ∧ (𝐴𝑅𝐵𝐵𝑅𝐶)) → 𝐵 ∈ V)
7 simpr 107 . . . . 5 ((𝜑 ∧ (𝐴𝑅𝐵𝐵𝑅𝐶)) → (𝐴𝑅𝐵𝐵𝑅𝐶))
8 breq2 3793 . . . . . . 7 (𝑥 = 𝐵 → (𝐴𝑅𝑥𝐴𝑅𝐵))
9 breq1 3792 . . . . . . 7 (𝑥 = 𝐵 → (𝑥𝑅𝐶𝐵𝑅𝐶))
108, 9anbi12d 450 . . . . . 6 (𝑥 = 𝐵 → ((𝐴𝑅𝑥𝑥𝑅𝐶) ↔ (𝐴𝑅𝐵𝐵𝑅𝐶)))
1110spcegv 2656 . . . . 5 (𝐵 ∈ V → ((𝐴𝑅𝐵𝐵𝑅𝐶) → ∃𝑥(𝐴𝑅𝑥𝑥𝑅𝐶)))
126, 7, 11sylc 60 . . . 4 ((𝜑 ∧ (𝐴𝑅𝐵𝐵𝑅𝐶)) → ∃𝑥(𝐴𝑅𝑥𝑥𝑅𝐶))
13 simpl 106 . . . . . 6 ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐵)
14 brrelex 4407 . . . . . 6 ((Rel 𝑅𝐴𝑅𝐵) → 𝐴 ∈ V)
153, 13, 14syl2an 277 . . . . 5 ((𝜑 ∧ (𝐴𝑅𝐵𝐵𝑅𝐶)) → 𝐴 ∈ V)
16 brrelex2 4408 . . . . . 6 ((Rel 𝑅𝐵𝑅𝐶) → 𝐶 ∈ V)
173, 4, 16syl2an 277 . . . . 5 ((𝜑 ∧ (𝐴𝑅𝐵𝐵𝑅𝐶)) → 𝐶 ∈ V)
18 brcog 4527 . . . . 5 ((𝐴 ∈ V ∧ 𝐶 ∈ V) → (𝐴(𝑅𝑅)𝐶 ↔ ∃𝑥(𝐴𝑅𝑥𝑥𝑅𝐶)))
1915, 17, 18syl2anc 397 . . . 4 ((𝜑 ∧ (𝐴𝑅𝐵𝐵𝑅𝐶)) → (𝐴(𝑅𝑅)𝐶 ↔ ∃𝑥(𝐴𝑅𝑥𝑥𝑅𝐶)))
2012, 19mpbird 160 . . 3 ((𝜑 ∧ (𝐴𝑅𝐵𝐵𝑅𝐶)) → 𝐴(𝑅𝑅)𝐶)
2120ex 112 . 2 (𝜑 → ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴(𝑅𝑅)𝐶))
22 df-er 6134 . . . . . 6 (𝑅 Er 𝑋 ↔ (Rel 𝑅 ∧ dom 𝑅 = 𝑋 ∧ (𝑅 ∪ (𝑅𝑅)) ⊆ 𝑅))
2322simp3bi 930 . . . . 5 (𝑅 Er 𝑋 → (𝑅 ∪ (𝑅𝑅)) ⊆ 𝑅)
241, 23syl 14 . . . 4 (𝜑 → (𝑅 ∪ (𝑅𝑅)) ⊆ 𝑅)
2524unssbd 3146 . . 3 (𝜑 → (𝑅𝑅) ⊆ 𝑅)
2625ssbrd 3830 . 2 (𝜑 → (𝐴(𝑅𝑅)𝐶𝐴𝑅𝐶))
2721, 26syld 44 1 (𝜑 → ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102   = wceq 1257  wex 1395  wcel 1407  Vcvv 2572  cun 2940  wss 2942   class class class wbr 3789  ccnv 4369  dom cdm 4370  ccom 4374  Rel wrel 4375   Er wer 6131
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-bndl 1413  ax-4 1414  ax-14 1419  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036  ax-sep 3900  ax-pow 3952  ax-pr 3969
This theorem depends on definitions:  df-bi 114  df-3an 896  df-tru 1260  df-nf 1364  df-sb 1660  df-eu 1917  df-mo 1918  df-clab 2041  df-cleq 2047  df-clel 2050  df-nfc 2181  df-ral 2326  df-rex 2327  df-v 2574  df-un 2947  df-in 2949  df-ss 2956  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-br 3790  df-opab 3844  df-xp 4376  df-rel 4377  df-co 4379  df-er 6134
This theorem is referenced by:  ertrd  6150  erth  6178  iinerm  6206  entr  6292
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