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Theorem iseriALT 8500
Description: Alternate proof of iseri 8499, avoiding the usage of mptru 1548 and as antecedent by using ax-mp 5 and one of the hypotheses as antecedent. This results, however, in a slightly longer proof. (Contributed by AV, 30-Apr-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
iseri.1 Rel 𝑅
iseri.2 (𝑥𝑅𝑦𝑦𝑅𝑥)
iseri.3 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)
iseri.4 (𝑥𝐴𝑥𝑅𝑥)
Assertion
Ref Expression
iseriALT 𝑅 Er 𝐴
Distinct variable groups:   𝑥,𝑦,𝑧,𝑅   𝑥,𝐴
Allowed substitution hints:   𝐴(𝑦,𝑧)

Proof of Theorem iseriALT
StepHypRef Expression
1 iseri.1 . 2 Rel 𝑅
2 id 22 . . 3 (Rel 𝑅 → Rel 𝑅)
3 iseri.2 . . . 4 (𝑥𝑅𝑦𝑦𝑅𝑥)
43adantl 481 . . 3 ((Rel 𝑅𝑥𝑅𝑦) → 𝑦𝑅𝑥)
5 iseri.3 . . . 4 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)
65adantl 481 . . 3 ((Rel 𝑅 ∧ (𝑥𝑅𝑦𝑦𝑅𝑧)) → 𝑥𝑅𝑧)
7 iseri.4 . . . 4 (𝑥𝐴𝑥𝑅𝑥)
87a1i 11 . . 3 (Rel 𝑅 → (𝑥𝐴𝑥𝑅𝑥))
92, 4, 6, 8iserd 8498 . 2 (Rel 𝑅𝑅 Er 𝐴)
101, 9ax-mp 5 1 𝑅 Er 𝐴
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  wcel 2109   class class class wbr 5078  Rel wrel 5593   Er wer 8469
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-11 2157  ax-12 2174  ax-ext 2710  ax-sep 5226  ax-nul 5233  ax-pr 5355
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-sb 2071  df-clab 2717  df-cleq 2731  df-clel 2817  df-ral 3070  df-rex 3071  df-rab 3074  df-v 3432  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-sn 4567  df-pr 4569  df-op 4573  df-br 5079  df-opab 5141  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-er 8472
This theorem is referenced by: (None)
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