0nelop - Intuitionistic Logic Explorer

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Theorem 0nelop 4233

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 3789	. . . . 5 $/\$
3		dfopg 3763	. . . . 5 $/\$
4	2, 3	syl 14	. . . 4
5	1, 4	eleqtrd 2249	. . 3
6		elpri 3606	. . 3 $\/$
7	5, 6	syl 14	. 2 $\/$
8	2	simpld 111	. . . . . 6
9		snnzg 3700	. . . . . 6
10	8, 9	syl 14	. . . . 5
11	10	necomd 2426	. . . 4
12		prnzg 3707	. . . . . 6
13	8, 12	syl 14	. . . . 5
14	13	necomd 2426	. . . 4
15	11, 14	jca 304	. . 3 $/\$
16		neanior 2427	. . 3 $/\$ $\/$
17	15, 16	sylib 121	. 2 $\/$
18	7, 17	pm2.65i 634	1

Colors of variables: wff set class

Syntax hints:

wn 3 $/\$ wa 103 $\/$ wo 703

wceq 1348

wcel 2141

wne 2340

cvv 2730

c0 3414

csn 3583

cpr 3584

cop 3586

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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152

This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-v 2732 df-dif 3123 df-un 3125 df-nul 3415 df-sn 3589 df-pr 3590 df-op 3592

This theorem is referenced by: 0nelelxp 4640