0neqopab - Intuitionistic Logic Explorer

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Theorem 0neqopab 5887

Description: The empty set is never an element in an ordered-pair class abstraction. (Contributed by Alexander van der Vekens, 5-Nov-2017.)

**Assertion**
Ref	Expression
0neqopab

**Proof of Theorem 0neqopab**
Step	Hyp	Ref	Expression
1		id 19	. 2
2		elopab 4236	. . 3 $/\$
3		nfopab1 4051	. . . . . 6
4	3	nfel2 2321	. . . . 5
5	4	nfn 1646	. . . 4
6		nfopab2 4052	. . . . . . 7
7	6	nfel2 2321	. . . . . 6
8	7	nfn 1646	. . . . 5
9		vex 2729	. . . . . . . 8
10		vex 2729	. . . . . . . 8
11	9, 10	opnzi 4213	. . . . . . 7
12		nesym 2381	. . . . . . . 8
13		pm2.21 607	. . . . . . . 8
14	12, 13	sylbi 120	. . . . . . 7
15	11, 14	ax-mp 5	. . . . . 6
16	15	adantr 274	. . . . 5 $/\$
17	8, 16	exlimi 1582	. . . 4 $/\$
18	5, 17	exlimi 1582	. . 3 $/\$
19	2, 18	sylbi 120	. 2
20	1, 19	pm2.65i 629	1

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 103

wceq 1343

wex 1480

wcel 2136

wne 2336

c0 3409

cop 3579

copab 4042

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-14 2139 ax-ext 2147 ax-sep 4100 ax-pow 4153 ax-pr 4187

This theorem depends on definitions: df-bi 116 df-3an 970 df-tru 1346 df-fal 1349 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-in 3122 df-ss 3129 df-nul 3410 df-pw 3561 df-sn 3582 df-pr 3583 df-op 3585 df-opab 4044

This theorem is referenced by: (None)