iseri - Metamath Proof Explorer

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Theorem iseri 7769

Description: A reflexive, symmetric, transitive relation is an equivalence relation on its domain. Inference version of iserd 7768, which avoids the need to provide a "dummy antecedent"

if there is no natural one to choose. (Contributed by AV, 30-Apr-2021.)

**Hypotheses**
Ref	Expression
iseri.1
iseri.2
iseri.3	$/\$
iseri.4

**Assertion**
Ref	Expression
iseri

Distinct variable groups:

Allowed substitution hints:

(

)

**Proof of Theorem iseri**
Step	Hyp	Ref	Expression
1		iseri.1	. . . 4
2	1	a1i 11	. . 3
3		iseri.2	. . . 4
4	3	adantl 482	. . 3 $/\$
5		iseri.3	. . . 4 $/\$
6	5	adantl 482	. . 3 $/\$ $/\$
7		iseri.4	. . . 4
8	7	a1i 11	. . 3
9	2, 4, 6, 8	iserd 7768	. 2
10	9	trud 1493	1

Colors of variables: wff setvar class

Syntax hints:

wi 4

wb 196 $/\$ wa 384

wtru 1484

wcel 1990 class class class wbr 4653

wrel 5119

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: eqer 7777 0er 7780 ecopover 7851 ener 8002 gicer 17718 phtpcer 22794 vitalilem1 23376 erclwwlks 26937 erclwwlksn 26949

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