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Theorem iseri 7922
Description: A reflexive, symmetric, transitive relation is an equivalence relation on its domain. Inference version of iserd 7921, 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 467 . . 3 ((⊤ ∧ 𝑥𝑅𝑦) → 𝑦𝑅𝑥)
5 iseri.3 . . . 4 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)
65adantl 467 . . 3 ((⊤ ∧ (𝑥𝑅𝑦𝑦𝑅𝑧)) → 𝑥𝑅𝑧)
7 iseri.4 . . . 4 (𝑥𝐴𝑥𝑅𝑥)
87a1i 11 . . 3 (⊤ → (𝑥𝐴𝑥𝑅𝑥))
92, 4, 6, 8iserd 7921 . 2 (⊤ → 𝑅 Er 𝐴)
109trud 1640 1 𝑅 Er 𝐴
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  wtru 1631  wcel 2144   class class class wbr 4784  Rel wrel 5254   Er wer 7892
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-br 4785  df-opab 4845  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-er 7895
This theorem is referenced by:  eqer  7930  0er  7932  ecopover  8003  ener  8155  gicer  17925  phtpcer  23013  vitalilem1  23595  erclwwlk  27170  erclwwlkn  27227
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