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Theorem ertr 8296
 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 8290 . . . . . . 7 (𝑅 Er 𝑋 → Rel 𝑅)
31, 2syl 17 . . . . . 6 (𝜑 → Rel 𝑅)
4 simpr 487 . . . . . 6 ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐵𝑅𝐶)
5 brrelex1 5598 . . . . . 6 ((Rel 𝑅𝐵𝑅𝐶) → 𝐵 ∈ V)
63, 4, 5syl2an 597 . . . . 5 ((𝜑 ∧ (𝐴𝑅𝐵𝐵𝑅𝐶)) → 𝐵 ∈ V)
7 simpr 487 . . . . 5 ((𝜑 ∧ (𝐴𝑅𝐵𝐵𝑅𝐶)) → (𝐴𝑅𝐵𝐵𝑅𝐶))
8 breq2 5061 . . . . . 6 (𝑥 = 𝐵 → (𝐴𝑅𝑥𝐴𝑅𝐵))
9 breq1 5060 . . . . . 6 (𝑥 = 𝐵 → (𝑥𝑅𝐶𝐵𝑅𝐶))
108, 9anbi12d 632 . . . . 5 (𝑥 = 𝐵 → ((𝐴𝑅𝑥𝑥𝑅𝐶) ↔ (𝐴𝑅𝐵𝐵𝑅𝐶)))
116, 7, 10spcedv 3597 . . . 4 ((𝜑 ∧ (𝐴𝑅𝐵𝐵𝑅𝐶)) → ∃𝑥(𝐴𝑅𝑥𝑥𝑅𝐶))
12 simpl 485 . . . . . 6 ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐵)
13 brrelex1 5598 . . . . . 6 ((Rel 𝑅𝐴𝑅𝐵) → 𝐴 ∈ V)
143, 12, 13syl2an 597 . . . . 5 ((𝜑 ∧ (𝐴𝑅𝐵𝐵𝑅𝐶)) → 𝐴 ∈ V)
15 brrelex2 5599 . . . . . 6 ((Rel 𝑅𝐵𝑅𝐶) → 𝐶 ∈ V)
163, 4, 15syl2an 597 . . . . 5 ((𝜑 ∧ (𝐴𝑅𝐵𝐵𝑅𝐶)) → 𝐶 ∈ V)
17 brcog 5730 . . . . 5 ((𝐴 ∈ V ∧ 𝐶 ∈ V) → (𝐴(𝑅𝑅)𝐶 ↔ ∃𝑥(𝐴𝑅𝑥𝑥𝑅𝐶)))
1814, 16, 17syl2anc 586 . . . 4 ((𝜑 ∧ (𝐴𝑅𝐵𝐵𝑅𝐶)) → (𝐴(𝑅𝑅)𝐶 ↔ ∃𝑥(𝐴𝑅𝑥𝑥𝑅𝐶)))
1911, 18mpbird 259 . . 3 ((𝜑 ∧ (𝐴𝑅𝐵𝐵𝑅𝐶)) → 𝐴(𝑅𝑅)𝐶)
2019ex 415 . 2 (𝜑 → ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴(𝑅𝑅)𝐶))
21 df-er 8281 . . . . . 6 (𝑅 Er 𝑋 ↔ (Rel 𝑅 ∧ dom 𝑅 = 𝑋 ∧ (𝑅 ∪ (𝑅𝑅)) ⊆ 𝑅))
2221simp3bi 1141 . . . . 5 (𝑅 Er 𝑋 → (𝑅 ∪ (𝑅𝑅)) ⊆ 𝑅)
231, 22syl 17 . . . 4 (𝜑 → (𝑅 ∪ (𝑅𝑅)) ⊆ 𝑅)
2423unssbd 4162 . . 3 (𝜑 → (𝑅𝑅) ⊆ 𝑅)
2524ssbrd 5100 . 2 (𝜑 → (𝐴(𝑅𝑅)𝐶𝐴𝑅𝐶))
2620, 25syld 47 1 (𝜑 → ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1530  ∃wex 1773   ∈ wcel 2107  Vcvv 3493   ∪ cun 3932   ⊆ wss 3934   class class class wbr 5057  ◡ccnv 5547  dom cdm 5548   ∘ ccom 5552  Rel wrel 5553   Er wer 8278 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-br 5058  df-opab 5120  df-xp 5554  df-rel 5555  df-co 5557  df-er 8281 This theorem is referenced by:  ertrd  8297  erth  8330  iiner  8361  entr  8553  efginvrel2  18845  efgsrel  18852
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