Theorem prneimg 4211

Description: Two pairs are not equal if at least one element of the first pair is not contained in the second pair. (Contributed by Alexander van der Vekens, 13-Aug-2017.)

Assertion

Ref	Expression
prneimg

Proof of Theorem prneimg

Step	Hyp	Ref	Expression
1		preq12bg 4209	. . . . 5
2		orddi 869	. . . . . 6
3		simpll 753	. . . . . . 7
4		pm1.4 386	. . . . . . . 8
5	4	ad2antll 728	. . . . . . 7
6	3, 5	jca 532	. . . . . 6
7	2, 6	sylbi 195	. . . . 5
8	1, 7	syl6bi 228	. . . 4
9		ianor 488	. . . . . 6
10		nne 2658	. . . . . . 7
11		nne 2658	. . . . . . 7
12	10, 11	orbi12i 521	. . . . . 6
13	9, 12	bitr2i 250	. . . . 5
14		ianor 488	. . . . . 6
15		nne 2658	. . . . . . 7
16		nne 2658	. . . . . . 7
17	15, 16	orbi12i 521	. . . . . 6
18	14, 17	bitr2i 250	. . . . 5
19	13, 18	anbi12i 697	. . . 4
20	8, 19	syl6ib 226	. . 3
21		pm4.56 495	. . 3
22	20, 21	syl6ib 226	. 2
23	22	necon2ad 2670	1

Syntax hints:  -.wn 3  ->wi 4  \/wo 368  /\wa 369  =wceq 1395  e.wcel 1818  =/=wne 2652  {cpr 4031

This theorem is referenced by:  prnebg  4212  symg2bas  16423  m2detleib  19133  usgraexmpldifpr  24400  usgvad2edg  32411  zlmodzxzldeplem  33099

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-v 3111  df-un 3480  df-sn 4030  df-pr 4032