Theorem iseriALT 8526

Description: Alternate proof of iseri 8525, avoiding the usage of mptru 1546 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	⊢ (𝑥𝑅𝑦 → 𝑦𝑅𝑥)
iseri.3	⊢ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)
iseri.4	⊢ (𝑥 ∈ 𝐴 ↔ 𝑥𝑅𝑥)

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

Step	Hyp	Ref	Expression
1		iseri.1	. 2 ⊢ Rel 𝑅
2		id 22	. . 3 ⊢ (Rel 𝑅 → Rel 𝑅)
3		iseri.2	. . . 4 ⊢ (𝑥𝑅𝑦 → 𝑦𝑅𝑥)
4	3	adantl 482	. . 3 ⊢ ((Rel 𝑅 ∧ 𝑥𝑅𝑦) → 𝑦𝑅𝑥)
5		iseri.3	. . . 4 ⊢ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)
6	5	adantl 482	. . 3 ⊢ ((Rel 𝑅 ∧ (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧)) → 𝑥𝑅𝑧)
7		iseri.4	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥𝑅𝑥)
8	7	a1i 11	. . 3 ⊢ (Rel 𝑅 → (𝑥 ∈ 𝐴 ↔ 𝑥𝑅𝑥))
9	2, 4, 6, 8	iserd 8524	. 2 ⊢ (Rel 𝑅 → 𝑅 Er 𝐴)
10	1, 9	ax-mp 5	1 ⊢ 𝑅 Er 𝐴

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∈ wcel 2106   class class class wbr 5074  Rel wrel 5594   Er wer 8495

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-11 2154  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-er 8498

This theorem is referenced by: (None)