0neqopab - Intuitionistic Logic Explorer

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Theorem 0neqopab 5963

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 4288	. . 3 $/\$
3		nfopab1 4098	. . . . . 6
4	3	nfel2 2349	. . . . 5
5	4	nfn 1669	. . . 4
6		nfopab2 4099	. . . . . . 7
7	6	nfel2 2349	. . . . . 6
8	7	nfn 1669	. . . . 5
9		vex 2763	. . . . . . . 8
10		vex 2763	. . . . . . . 8
11	9, 10	opnzi 4264	. . . . . . 7
12		nesym 2409	. . . . . . . 8
13		pm2.21 618	. . . . . . . 8
14	12, 13	sylbi 121	. . . . . . 7
15	11, 14	ax-mp 5	. . . . . 6
16	15	adantr 276	. . . . 5 $/\$
17	8, 16	exlimi 1605	. . . 4 $/\$
18	5, 17	exlimi 1605	. . 3 $/\$
19	2, 18	sylbi 121	. 2
20	1, 19	pm2.65i 640	1

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104

wceq 1364

wex 1503

wcel 2164

wne 2364

c0 3446

cop 3621

copab 4089

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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2167 ax-ext 2175 ax-sep 4147 ax-pow 4203 ax-pr 4238

This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-v 2762 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166 df-nul 3447 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-opab 4091

This theorem is referenced by: (None)