Theorem efrirr 2928

Description: Irreflexivitiy of the epsilon relation: a class founded by epsilon is not a member of itself.

Assertion

Ref	Expression
efrirr	|- (E Fr A -> -. A e. A)

Proof of Theorem efrirr

Step	Hyp	Ref	Expression
1		freq2 2923	. . . . 5 |- (x = A -> (E Fr x <-> E Fr A))
2		eleq1 1534	. . . . . . 7 |- (x = A -> (x e. x <-> A e. x))
3		eleq2 1535	. . . . . . 7 |- (x = A -> (A e. x <-> A e. A))
4	2, 3	bitrd 528	. . . . . 6 |- (x = A -> (x e. x <-> A e. A))
5	4	negbid 611	. . . . 5 |- (x = A -> (-. x e. x <-> -. A e. A))
6	1, 5	imbi12d 626	. . . 4 |- (x = A -> ((E Fr x -> -. x e. x) <-> (E Fr A -> -. A e. A)))
7		frirr 2924	. . . . . . 7 |- ((E Fr x /\ x e. x) -> -. xEx)
8	7	ex 373	. . . . . 6 |- (E Fr x -> (x e. x -> -. xEx))
9		epel 2834	. . . . . . 7 |- (xEx <-> x e. x)
10	9	negbii 187	. . . . . 6 |- (-. xEx <-> -. x e. x)
11	8, 10	syl6ib 212	. . . . 5 |- (E Fr x -> (x e. x -> -. x e. x))
12	11	pm2.01d 89	. . . 4 |- (E Fr x -> -. x e. x)
13	6, 12	vtoclg 1847	. . 3 |- (A e. A -> (E Fr A -> -. A e. A))
14	13	com12 11	. 2 |- (E Fr A -> (A e. A -> -. A e. A))
15	14	pm2.01d 89	1 |- (E Fr A -> -. A e. A)

Syntax hints:  -. wn 2   -> wi 3   = wceq 956   e. wcel 958   class class class wbr 2619  Ecep 2830   Fr wfr 2915

This theorem is referenced by:  tz7.2 2931  ordirr 2966

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-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-3an 777  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-ral 1649  df-rex 1650  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-eprel 2832  df-fr 2917

Copyright terms: Public domain