0nelop - Intuitionistic Logic Explorer

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Theorem 0nelop 4130

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 3695	. . . . 5 $/\$
3		dfopg 3669	. . . . 5 $/\$
4	2, 3	syl 14	. . . 4
5	1, 4	eleqtrd 2193	. . 3
6		elpri 3516	. . 3 $\/$
7	5, 6	syl 14	. 2 $\/$
8	2	simpld 111	. . . . . 6
9		snnzg 3606	. . . . . 6
10	8, 9	syl 14	. . . . 5
11	10	necomd 2368	. . . 4
12		prnzg 3613	. . . . . 6
13	8, 12	syl 14	. . . . 5
14	13	necomd 2368	. . . 4
15	11, 14	jca 302	. . 3 $/\$
16		neanior 2369	. . 3 $/\$ $\/$
17	15, 16	sylib 121	. 2 $\/$
18	7, 17	pm2.65i 611	1

Colors of variables: wff set class

Syntax hints:

wn 3 $/\$ wa 103 $\/$ wo 680

wceq 1314

wcel 1463

wne 2282

cvv 2657

c0 3329

csn 3493

cpr 3494

cop 3496

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 586 ax-in2 587 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097

This theorem depends on definitions: df-bi 116 df-3an 947 df-tru 1317 df-nf 1420 df-sb 1719 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-ne 2283 df-v 2659 df-dif 3039 df-un 3041 df-nul 3330 df-sn 3499 df-pr 3500 df-op 3502

This theorem is referenced by: 0nelelxp 4528