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Theorem iseri 7656
 Description: A reflexive, symmetric, transitive relation is an equivalence relation on its domain. Inference version of iserd 7655, which avoids the need to provide a "dummy antecedent" 𝜑 if there is no natural one to choose. (Contributed by AV, 30-Apr-2021.)
Hypotheses
Ref Expression
iseri.1 Rel 𝑅
iseri.2 (𝑥𝑅𝑦𝑦𝑅𝑥)
iseri.3 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)
iseri.4 (𝑥𝐴𝑥𝑅𝑥)
Assertion
Ref Expression
iseri 𝑅 Er 𝐴
Distinct variable groups:   𝑥,𝑦,𝑧,𝑅   𝑥,𝐴
Allowed substitution hints:   𝐴(𝑦,𝑧)

Proof of Theorem iseri
StepHypRef Expression
1 iseri.1 . . . 4 Rel 𝑅
21a1i 11 . . 3 (⊤ → Rel 𝑅)
3 iseri.2 . . . 4 (𝑥𝑅𝑦𝑦𝑅𝑥)
43adantl 481 . . 3 ((⊤ ∧ 𝑥𝑅𝑦) → 𝑦𝑅𝑥)
5 iseri.3 . . . 4 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)
65adantl 481 . . 3 ((⊤ ∧ (𝑥𝑅𝑦𝑦𝑅𝑧)) → 𝑥𝑅𝑧)
7 iseri.4 . . . 4 (𝑥𝐴𝑥𝑅𝑥)
87a1i 11 . . 3 (⊤ → (𝑥𝐴𝑥𝑅𝑥))
92, 4, 6, 8iserd 7655 . 2 (⊤ → 𝑅 Er 𝐴)
109trud 1484 1 𝑅 Er 𝐴
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  ⊤wtru 1476   ∈ wcel 1977   class class class wbr 4583  Rel wrel 5043   Er wer 7626 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-er 7629 This theorem is referenced by:  eqer  7664  0er  7667  ecopover  7738  ener  7888  gicer  17541  phtpcer  22602  vitalilem1  23182  erclwwlk  26344  erclwwlkn  26356  erclwwlks  41244  erclwwlksn  41256
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