prneimg - Metamath Proof Explorer

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Theorem prneimg 4388

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 4386	. . . . 5 $/\$ $/\$ $/\$ $/\$ $\/$ $/\$
2		orddi 913	. . . . . 6 $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$
3		simpll 790	. . . . . . 7 $\/$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $\/$
4		pm1.4 401	. . . . . . . 8 $\/$ $\/$
5	4	ad2antll 765	. . . . . . 7 $\/$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $\/$
6	3, 5	jca 554	. . . . . 6 $\/$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $\/$ $/\$ $\/$
7	2, 6	sylbi 207	. . . . 5 $/\$ $\/$ $/\$ $\/$ $/\$ $\/$
8	1, 7	syl6bi 243	. . . 4 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$
9		ianor 509	. . . . . 6 $/\$ $\/$
10		nne 2798	. . . . . . 7
11		nne 2798	. . . . . . 7
12	10, 11	orbi12i 543	. . . . . 6 $\/$ $\/$
13	9, 12	bitr2i 265	. . . . 5 $\/$ $/\$
14		ianor 509	. . . . . 6 $/\$ $\/$
15		nne 2798	. . . . . . 7
16		nne 2798	. . . . . . 7
17	15, 16	orbi12i 543	. . . . . 6 $\/$ $\/$
18	14, 17	bitr2i 265	. . . . 5 $\/$ $/\$
19	13, 18	anbi12i 733	. . . 4 $\/$ $/\$ $\/$ $/\$ $/\$ $/\$
20	8, 19	syl6ib 241	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
21		pm4.56 516	. . 3 $/\$ $/\$ $/\$ $/\$ $\/$ $/\$
22	20, 21	syl6ib 241	. 2 $/\$ $/\$ $/\$ $/\$ $\/$ $/\$
23	22	necon2ad 2809	1 $/\$ $/\$ $/\$ $/\$ $\/$ $/\$

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4 $\/$ wo 383 $/\$ wa 384

wceq 1483

wcel 1990

wne 2794

cpr 4179

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

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-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-v 3202 df-un 3579 df-sn 4178 df-pr 4180

This theorem is referenced by: prnebg 4389 symg2bas 17818 m2detleib 20437 umgrvad2edg 26105 usgrexmpldifpr 26150 zlmodzxzldeplem 42287

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