0nelop - Intuitionistic Logic Explorer

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Theorem 0nelop 3985

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 3573	. . . . 5 $/\$
3		dfopg 3547	. . . . 5 $/\$
4	2, 3	syl 14	. . . 4
5	1, 4	eleqtrd 2116	. . 3
6		elpri 3398	. . 3 $\/$
7	5, 6	syl 14	. 2 $\/$
8	2	simpld 105	. . . . . 6
9		snnzg 3485	. . . . . 6
10	8, 9	syl 14	. . . . 5
11	10	necomd 2291	. . . 4
12		prnzg 3492	. . . . . 6
13	8, 12	syl 14	. . . . 5
14	13	necomd 2291	. . . 4
15	11, 14	jca 290	. . 3 $/\$
16		neanior 2292	. . 3 $/\$ $\/$
17	15, 16	sylib 127	. 2 $\/$
18	7, 17	pm2.65i 568	1

Colors of variables: wff set class

Syntax hints:

wn 3 $/\$ wa 97 $\/$ wo 629

wceq 1243

wcel 1393

wne 2204

cvv 2557

c0 3224

csn 3375

cpr 3376

cop 3378

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022

This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-v 2559 df-dif 2920 df-un 2922 df-nul 3225 df-sn 3381 df-pr 3382 df-op 3384

This theorem is referenced by: 0nelelxp 4373