Theorem iseriALT 7770

Description: Alternate proof of iseri 7769, avoiding the usage of trud 1493 and T. 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	|- ( x R y -> y R x )
iseri.3	|- ( ( x R y /\ y R z ) -> x R z )
iseri.4	|- ( x e. A <-> x R x )

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 |- ( x R y -> y R x )
4	3	adantl 482	. . 3 |- ( ( Rel R /\ x R y ) -> y R x )
5		iseri.3	. . . 4 |- ( ( x R y /\ y R z ) -> x R z )
6	5	adantl 482	. . 3 |- ( ( Rel R /\ ( x R y /\ y R z ) ) -> x R z )
7		iseri.4	. . . 4 |- ( x e. A <-> x R x )
8	7	a1i 11	. . 3 |- ( Rel R -> ( x e. A <-> x R x ) )
9	2, 4, 6, 8	iserd 7768	. 2 |- ( Rel R -> R Er A )
10	1, 9	ax-mp 5	1 |- R Er A

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   e. wcel 1990   class class class wbr 4653   Rel wrel 5119   Er wer 7739

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-er 7742

This theorem is referenced by: (None)