0nelop - Intuitionistic Logic Explorer

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Theorem 0nelop 4226

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 3782	. . . . 5 $/\$
3		dfopg 3756	. . . . 5 $/\$
4	2, 3	syl 14	. . . 4
5	1, 4	eleqtrd 2245	. . 3
6		elpri 3599	. . 3 $\/$
7	5, 6	syl 14	. 2 $\/$
8	2	simpld 111	. . . . . 6
9		snnzg 3693	. . . . . 6
10	8, 9	syl 14	. . . . 5
11	10	necomd 2422	. . . 4
12		prnzg 3700	. . . . . 6
13	8, 12	syl 14	. . . . 5
14	13	necomd 2422	. . . 4
15	11, 14	jca 304	. . 3 $/\$
16		neanior 2423	. . 3 $/\$ $\/$
17	15, 16	sylib 121	. 2 $\/$
18	7, 17	pm2.65i 629	1

Colors of variables: wff set class

Syntax hints:

wn 3 $/\$ wa 103 $\/$ wo 698

wceq 1343

wcel 2136

wne 2336

cvv 2726

c0 3409

csn 3576

cpr 3577

cop 3579

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 604 ax-in2 605 ax-io 699 ax-5 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-ext 2147

This theorem depends on definitions: df-bi 116 df-3an 970 df-tru 1346 df-nf 1449 df-sb 1751 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-ne 2337 df-v 2728 df-dif 3118 df-un 3120 df-nul 3410 df-sn 3582 df-pr 3583 df-op 3585

This theorem is referenced by: 0nelelxp 4633