Theorem ider 4269

Description: The identity relation is an equivalence relation.

Assertion

Ref	Expression
ider	|- Er I

Proof of Theorem ider

Step	Hyp	Ref	Expression
1		equcomi 1128	. . 3 |- (x = y -> y = x)
2		visset 1813	. . . 4 |- y e. V
3	2	ideq 3277	. . 3 |- (xIy <-> x = y)
4		visset 1813	. . . 4 |- x e. V
5	4	ideq 3277	. . 3 |- (yIx <-> y = x)
6	1, 3, 5	3imtr4 219	. 2 |- (xIy -> yIx)
7		eqtrt 1492	. . 3 |- ((x = y /\ y = z) -> x = z)
8		visset 1813	. . . . 5 |- z e. V
9	8	ideq 3277	. . . 4 |- (yIz <-> y = z)
10	3, 9	anbi12i 482	. . 3 |- ((xIy /\ yIz) <-> (x = y /\ y = z))
11	8	ideq 3277	. . 3 |- (xIz <-> x = z)
12	7, 10, 11	3imtr4 219	. 2 |- ((xIy /\ yIz) -> xIz)
13	6, 12	ster 4268	1 |- Er I

Syntax hints:   /\ wa 223   = wceq 956   class class class wbr 2619  Icid 2831  Er wer 4258

This theorem is referenced by:  erdisj2 10442

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-er 4261

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