0neqopab - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > 0neqopab		Unicode version

Theorem 0neqopab 6049

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 4346	. . 3 $/\$
3		nfopab1 4153	. . . . . 6
4	3	nfel2 2385	. . . . 5
5	4	nfn 1704	. . . 4
6		nfopab2 4154	. . . . . . 7
7	6	nfel2 2385	. . . . . 6
8	7	nfn 1704	. . . . 5
9		vex 2802	. . . . . . . 8
10		vex 2802	. . . . . . . 8
11	9, 10	opnzi 4321	. . . . . . 7
12		nesym 2445	. . . . . . . 8
13		pm2.21 620	. . . . . . . 8
14	12, 13	sylbi 121	. . . . . . 7
15	11, 14	ax-mp 5	. . . . . 6
16	15	adantr 276	. . . . 5 $/\$
17	8, 16	exlimi 1640	. . . 4 $/\$
18	5, 17	exlimi 1640	. . 3 $/\$
19	2, 18	sylbi 121	. 2
20	1, 19	pm2.65i 642	1

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104

wceq 1395

wex 1538

wcel 2200

wne 2400

c0 3491

cop 3669

copab 4144

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-14 2203 ax-ext 2211 ax-sep 4202 ax-pow 4258 ax-pr 4293

This theorem depends on definitions: df-bi 117 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-v 2801 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-nul 3492 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-opab 4146

This theorem is referenced by: (None)