0nelop - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > 0nelop		Unicode version

Theorem 0nelop 4165

Description: A property of ordered pairs. (Contributed by Mario Carneiro, 26-Apr-2015.)

**Assertion**
Ref	Expression
0nelop

**Proof of Theorem 0nelop**
Step	Hyp	Ref	Expression
1		id 19	. . . 4
2		oprcl 3724	. . . . 5 $/\$
3		dfopg 3698	. . . . 5 $/\$
4	2, 3	syl 14	. . . 4
5	1, 4	eleqtrd 2216	. . 3
6		elpri 3545	. . 3 $\/$
7	5, 6	syl 14	. 2 $\/$
8	2	simpld 111	. . . . . 6
9		snnzg 3635	. . . . . 6
10	8, 9	syl 14	. . . . 5
11	10	necomd 2392	. . . 4
12		prnzg 3642	. . . . . 6
13	8, 12	syl 14	. . . . 5
14	13	necomd 2392	. . . 4
15	11, 14	jca 304	. . 3 $/\$
16		neanior 2393	. . 3 $/\$ $\/$
17	15, 16	sylib 121	. 2 $\/$
18	7, 17	pm2.65i 628	1

Colors of variables: wff set class

Syntax hints:

wn 3 $/\$ wa 103 $\/$ wo 697

wceq 1331

wcel 1480

wne 2306

cvv 2681

c0 3358

csn 3522

cpr 3523

cop 3525

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119

This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-v 2683 df-dif 3068 df-un 3070 df-nul 3359 df-sn 3528 df-pr 3529 df-op 3531

This theorem is referenced by: 0nelelxp 4563